pacman::p_load(sf, spatstat, raster, tidyverse, tmap, spNetwork, spdep, knitr)
set.seed(123)Take Home Exercise 1
1 Introduction
According to the World Health Organisation (WHO), road traffic accidents cause approximately 1.19 million deaths annually and result in between 20 and 50 million people with being down with non-fatal injuries. Over half of all road traffic deaths occur among vulnerable road users, such as pedestrians, cyclists and motorcyclists.
Road traffic injuries are the leading cause of death for children and young adults aged 5–29 while two-thirds of road traffic fatalities occur among people of working age (18–59 years). 9 in 10 fatalities on the roads occur in low and middle-income countries, even though these countries have only around 60% of the world’s vehicles.
In addition to the human suffering caused by road traffic injuries, they also inflict a heavy economic burden on victims and their families, both through treatment costs for the injured and through loss of productivity of those killed or disabled. More broadly, road traffic injuries have a serious impact on national economies, costing countries 3% of their annual GDP.
Thailand’s roads are the deadliest in Southeast Asia and among the worst in the world, according to the World Health Organisation. About 20,000 people die in road accidents each year, or about 56 deaths a day (WHO).
Between 2014 and 2021, Thailand experienced a notable increase in accidents. Specifically, 19% of all accidents in Thailand occurred on the national highways, which constituted the primary public thoroughfares connecting various regions, provinces, districts, and significant locations within a comprehensive network.
Within the broader context of accidents across the country, there existed a considerable 66% likelihood of encountering accident-prone zones, often termed ‘black spots,’ distributed as follows: 66% on straight road segments, 13% at curves, 6% at median points of cross-shaped intersections, 5% at T-shaped intersections and Y-shaped intersections, 3% at cross-shaped intersections, 2% on bridges, and 2% on steep slopes, respectively.
1.1 Objectives
For the most part, road accidents can be attributed to two key factors- behavioral factors. such as driving skill, and environmental factors, such as heavy rain.
For this exercise, we will implement Spatial and Spatio-Temporal Point Pattern Analysis methods to discover the underlying factors behind the accidents in the Bangkok Metropolitan Region.
1.2 Data and Packages
For the purpose of this study, we will use the following three data-frames.
Thailand Road Accidents [2019-2022], sourced from Kaggle.
Thailand Roads Open Street Map Export, sourced from Humanitarian Data Exchange.
Thailand Subnational Administrative Boundaries sourced from Humanitarian Data Exchange.
You can click the link embedded on each of these data-frames to learn more.
We will use the following R packages for our analysis:
sf, a relatively new R package specially designed to import, manage and process vector-based geospatial data in R.spatstat, which has a wide range of useful functions for point pattern analysis. In this hands-on exercise, it will be used to perform 1st- and 2nd-order spatial point patterns analysis and derive kernel density estimation (KDE) layer.raster, which reads, writes, manipulates, analyses and model of gridded spatial data (i.e. raster). In this hands-on exercise, it will be used to convert image output generate by spatstat into raster format.tidyversesimplifies spatial analysis by offering a consistent and efficient framework that facilitates working with spatial data.tmapwhich provides functions for plotting cartographic quality static point patterns maps or interactive maps by using leaflet API.spNetworkwhich provides functions to perform Spatial Point Patterns Analysis such as kernel density estimation (KDE) and K-function on network. It also can be used to build spatial matrices (‘listw’ objects like in ‘spdep’ package) to conduct any kind of traditional spatial analysis with spatial weights based on reticular distances.
We import them into our environment using the code chunk below.
Click to show/hide code chunk
rdacc_sf=read_rds("data/rds/acc_sf")%>%
mutate(hourofday= hour(incident_datetime))%>%
mutate(traffic_period = case_when(
# Define peak hours: 7-9 AM and 4-7 PM (16-19 in 24-hour format)
hourofday >= 7 & hourofday <= 9 ~ "Peak",
hourofday >= 16 & hourofday <= 19 ~ "Peak",
TRUE ~ "Off-Peak" # Everything else is off-peak
))
st_crs(rdacc_sf)Coordinate Reference System:
User input: EPSG:32647
wkt:
PROJCRS["WGS 84 / UTM zone 47N",
BASEGEOGCRS["WGS 84",
ENSEMBLE["World Geodetic System 1984 ensemble",
MEMBER["World Geodetic System 1984 (Transit)"],
MEMBER["World Geodetic System 1984 (G730)"],
MEMBER["World Geodetic System 1984 (G873)"],
MEMBER["World Geodetic System 1984 (G1150)"],
MEMBER["World Geodetic System 1984 (G1674)"],
MEMBER["World Geodetic System 1984 (G1762)"],
MEMBER["World Geodetic System 1984 (G2139)"],
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]],
ENSEMBLEACCURACY[2.0]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4326]],
CONVERSION["UTM zone 47N",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",0,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",99,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",0.9996,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",500000,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",0,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["(E)",east,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["(N)",north,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Navigation and medium accuracy spatial referencing."],
AREA["Between 96°E and 102°E, northern hemisphere between equator and 84°N, onshore and offshore. China. Indonesia. Laos. Malaysia - West Malaysia. Mongolia. Myanmar (Burma). Russian Federation. Thailand."],
BBOX[0,96,84,102]],
ID["EPSG",32647]]road_sf <- st_read(dsn = "data/rawdata", layer="hotosm_tha_roads_lines_shp")map_sf <- st_read(dsn = "data/rawdata", layer="tha_admbnda_adm1_rtsd_20220121")Reading layer `tha_admbnda_adm1_rtsd_20220121' from data source
`C:\arjxn11\ISSS626-GAA\Take-home_Ex\Take-home_Ex1\data\rawdata'
using driver `ESRI Shapefile'
Simple feature collection with 77 features and 16 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 97.34336 ymin: 5.613038 xmax: 105.637 ymax: 20.46507
Geodetic CRS: WGS 84map2_sf <- st_read(dsn = "data/rawdata", layer="tha_admbnda_adm2_rtsd_20220121")Reading layer `tha_admbnda_adm2_rtsd_20220121' from data source
`C:\arjxn11\ISSS626-GAA\Take-home_Ex\Take-home_Ex1\data\rawdata'
using driver `ESRI Shapefile'
Simple feature collection with 928 features and 19 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 97.34336 ymin: 5.613038 xmax: 105.637 ymax: 20.46507
Geodetic CRS: WGS 84map_sf=st_transform(map_sf,crs = 32647)
map2_sf=st_transform(map2_sf,crs = 32647)road_sf=st_set_crs(road_sf, 4326)st_crs(map_sf)Coordinate Reference System:
User input: EPSG:32647
wkt:
PROJCRS["WGS 84 / UTM zone 47N",
BASEGEOGCRS["WGS 84",
ENSEMBLE["World Geodetic System 1984 ensemble",
MEMBER["World Geodetic System 1984 (Transit)"],
MEMBER["World Geodetic System 1984 (G730)"],
MEMBER["World Geodetic System 1984 (G873)"],
MEMBER["World Geodetic System 1984 (G1150)"],
MEMBER["World Geodetic System 1984 (G1674)"],
MEMBER["World Geodetic System 1984 (G1762)"],
MEMBER["World Geodetic System 1984 (G2139)"],
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]],
ENSEMBLEACCURACY[2.0]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4326]],
CONVERSION["UTM zone 47N",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",0,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",99,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",0.9996,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",500000,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",0,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["(E)",east,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["(N)",north,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Navigation and medium accuracy spatial referencing."],
AREA["Between 96°E and 102°E, northern hemisphere between equator and 84°N, onshore and offshore. China. Indonesia. Laos. Malaysia - West Malaysia. Mongolia. Myanmar (Burma). Russian Federation. Thailand."],
BBOX[0,96,84,102]],
ID["EPSG",32647]]bmr_province=c("Bangkok", "Samut Prakan", "Samut Sakhon", "Nonthaburi", "Nakhon Pathom", "Pathum Thani")bmr_boundary=map_sf %>%
filter(ADM1_EN %in% bmr_province)
bmr_boundary2=map2_sf %>%
filter(ADM1_EN %in% bmr_province)st_bbox(bmr_boundary) xmin ymin xmax ymax
587893.5 1484413.7 712440.5 1579076.3 st_crs(road_sf) # This is likely EPSG:4326 (WGS84, latitude/longitude)
st_crs(bmr_boundary) # This is likely EPSG:32647 (UTM Zone 47N)# If road_sf was originally in EPSG:4326, reproject it properly
road_sf <- st_transform(road_sf, crs = 32647)bmr_roads=st_intersection(map2_sf, road_sf)#write_rds(bmr_roads, "data/rds/bmrroads")bmr_roads=read_rds("data/rds/bmrroads")bmr_districts=st_intersection(map2_sf, bmr_roads)EDA
#Implementing the filter() function with the help of the list above.
bmr_accsf=rdacc_sf %>%
filter(province_en %in% bmr_province)
# Step 1: Summarize the number of hospitalizations per province
accident_summary <- bmr_accsf %>%
group_by(province_en) %>%
summarize(total_accidents = n())
# Step 2: Visualize using ggplot2
ggplot(accident_summary, aes(x = reorder(province_en, total_accidents), y = total_accidents)) +
geom_bar(stat = "identity", fill = "darkred") +
coord_flip() + # Flip coordinates for better readability if many provinces
labs(title = "Total Accidents by Province",
x = "Province",
y = "Number of Accidents") +
theme_minimal()
# Count the number of accidents in Peak and Off-Peak periods
accident_counts_summary <- bmr_accsf %>%
group_by(traffic_period) %>%
summarise(accident_count = n())
# Create a bar chart showing the number of accidents in Peak vs Off-Peak
ggplot(data = accident_counts_summary, aes(x = traffic_period, y = accident_count)) +
geom_bar(stat = "identity", fill = "steelblue") + # Bar chart with specified accident counts
labs(title = "Number of Accidents: Peak vs Off-Peak",
x = "Traffic Period",
y = "Number of Accidents") +
theme_minimal() +
theme(plot.title = element_text(hjust = 0.5)) # Center the title
cause_count <- bmr_accsf %>%
group_by(presumed_cause) %>%
summarise(count = n()) # Filter the top 5 causes
top_5_causes <- cause_count %>%
top_n(5, count)
# Create the bar chart showing only the top 5 presumed causes
ggplot(top_5_causes, aes(x = reorder(presumed_cause, -count), y = count)) +
geom_bar(stat = "identity", fill = "skyblue", color = "black") +
geom_text(aes(label = count), vjust = -0.5) + # Add labels to show the count on top of the bars
theme_minimal() +
labs(title = "Top 5 Presumed Causes of Accidents",
x = "Presumed Cause",
y = "Accident Count") +
theme(axis.text.x = element_text(angle = 45, hjust = 1)) 
# Set tmap to interactive view mode (optional for exploration)
tmap_mode("plot")
# Create the map with boundaries and accident points
tm_shape(bmr_boundary) +
tm_borders(col = "black") + # Show the boundaries
tm_shape(bmr_accsf) +
tm_dots(col = "red", size = 0.01) + # Show accident points as red dots
tm_layout(frame = FALSE, legend.outside = TRUE)
# Switch back to static mode after rendering (if needed)
tmap_mode("plot")# Create the map with your existing OSM data, BMR boundaries, and accident points
tm_shape(bmr_roads) + # Use your preloaded OSM data
tm_lines(col='gray', lwd = 1) +
tm_shape(bmr_boundary) +
tm_borders(col = "black", lwd = 1) + # Plot BMR boundary with black borders
tm_shape(bmr_accsf) +
tm_dots(col = "red", size = 0.01) + # Show accident points as red dots
tm_layout(frame = FALSE, legend.outside = TRUE)bmr_acc <- as_Spatial(bmr_accsf)
bmr <- as_Spatial(bmr_boundary)write_rds(osm_data, "data/rds/osm_data_spatial")osm_data=read_rds("data/rds/osm_data_spatial")bmracc_sp <- as(bmr_acc, "SpatialPoints")
boundary_sp <- as(bmr, "SpatialPolygons")# Load necessary libraries
library(sf)
# Create a square grid that covers the BMR boundary
grid_size <- 5000 # Size of the grid cells in meters (adjust as necessary)
bmr_grid <- st_make_grid(bmr_boundary, cellsize = grid_size, square = TRUE)
# Convert the grid into an sf object
bmr_grid_sf <- st_sf(geometry = bmr_grid)
# Clip the grid to the BMR boundary (to remove grid cells outside the boundary)
bmr_grid_sf <- st_intersection(bmr_grid_sf, bmr_boundary)# Create a hexagonal grid covering the BMR boundary
bmr_hex_grid <- st_make_grid(bmr_boundary, cellsize = grid_size, square = FALSE)
# Convert to an sf object and clip to the BMR boundary
bmr_hex_grid_sf <- st_sf(geometry = bmr_hex_grid)
bmr_hex_grid_sf <- st_intersection(bmr_hex_grid_sf, bmr_boundary)Now, we want to determine the road type that has the most accidents.
For this, we create a buffer first as accidents don’t always happen directly on the road but just off it using the st_buffer() function. A buffer of 5 meters is created.
After that, we can perform a spatial join using the st_join() function of the sf package.
# Create a small buffer around the accident points (e.g., 50 meters)
accidents_buffered = st_buffer(bmr_accsf, dist = 10)
# Perform the spatial join again with the buffered accident points
accidents_on_roads = st_join(accidents_buffered, bmr_roads, join = st_intersects)
head(accidents_on_roads)Simple feature collection with 6 features and 51 fields
Geometry type: POLYGON
Dimension: XY
Bounding box: xmin: 627002.3 ymin: 1531930 xmax: 655421.7 ymax: 1533391
Projected CRS: WGS 84 / UTM zone 47N
# A tibble: 6 × 52
acc_code incident_datetime report_datetime province_th province_en
<dbl> <dttm> <dttm> <chr> <chr>
1 571882 2019-01-01 02:25:00 2019-01-02 17:32:00 นครปฐม Nakhon Pathom
2 571882 2019-01-01 02:25:00 2019-01-02 17:32:00 นครปฐม Nakhon Pathom
3 571882 2019-01-01 02:25:00 2019-01-02 17:32:00 นครปฐม Nakhon Pathom
4 571882 2019-01-01 02:25:00 2019-01-02 17:32:00 นครปฐม Nakhon Pathom
5 600001 2019-01-01 03:00:00 2019-01-05 10:33:00 นนทบุรี Nonthaburi
6 600001 2019-01-01 03:00:00 2019-01-05 10:33:00 นนทบุรี Nonthaburi
# ℹ 47 more variables: agency <chr>, route <chr>, vehicle_type <chr>,
# presumed_cause <chr>, accident_type <chr>,
# number_of_vehicles_involved <dbl>, number_of_fatalities <dbl>,
# number_of_injuries <dbl>, weather_condition <chr>, road_description <chr>,
# slope_description <chr>, geometry <POLYGON [m]>, Month <dbl>,
# Month_fac <ord>, dayofmonth <int>, hourofday <int>, traffic_period <chr>,
# Shape_Leng <dbl>, Shape_Area <dbl>, ADM1_EN <chr>, ADM1_TH <chr>, …We now count the number of accidents on each highway type by implementing the code chunk below.
# Count accidents by road type
accidents_by_road_type = accidents_on_roads %>%
group_by(highway) %>% # Assuming the OSM roads data has a 'road_type' column
summarise(accident_count = n())
# View the summary
print(accidents_by_road_type)Simple feature collection with 20 features and 2 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 591267.5 ymin: 1486836 xmax: 710176.1 ymax: 1576530
Projected CRS: WGS 84 / UTM zone 47N
# A tibble: 20 × 3
highway accident_count geometry
<chr> <int> <MULTIPOLYGON [m]>
1 construction 1217 (((629789.3 1496006, 629789.1 1496005, 629788.…
2 cycleway 8 (((665707.4 1532883, 665707.3 1532882, 665707.…
3 footway 218 (((627989 1495786, 627988.8 1495786, 627988.6 …
4 motorway 5366 (((661514.4 1506189, 661514 1506189, 661513.7 …
5 motorway_link 939 (((664470.2 1506809, 664469.9 1506809, 664469.…
6 path 16 (((664563 1509817, 664562.5 1509817, 664562 15…
7 primary 2959 (((617714.5 1511678, 617714.2 1511678, 617713.…
8 primary_link 484 (((626402.2 1497350, 626401.9 1497350, 626401.…
9 residential 552 (((616749.7 1492543, 616749.4 1492542, 616749.…
10 secondary 2815 (((621676.4 1492259, 621676.1 1492259, 621675.…
11 secondary_link 329 (((637728 1498891, 637727.8 1498890, 637727.5 …
12 service 723 (((617727.2 1487698, 617726.9 1487698, 617726.…
13 steps 39 (((637789.9 1501257, 637789.8 1501256, 637789.…
14 tertiary 411 (((616694.7 1490301, 616694.5 1490301, 616694.…
15 tertiary_link 9 (((683966 1504996, 683965.9 1504995, 683965.8 …
16 track 4 (((682585.7 1518975, 682585.6 1518975, 682585.…
17 trunk 2897 (((603768.7 1525407, 603768.4 1525406, 603768 …
18 trunk_link 289 (((639960.7 1500837, 639960.5 1500837, 639960.…
19 unclassified 122 (((622943.8 1491550, 622943.6 1491550, 622943.…
20 <NA> 953 (((616483.2 1486841, 616482.9 1486841, 616482.…top_5_highway=accidents_by_road_type %>%
arrange(desc(accident_count))%>%
slice(1:5)We will now use the ggplot2 package to produce a barplot to facilitate visualization and understand what type of highways result in the most accidents.
library(ggplot2)
# Create bar plot
ggplot(top_5_highway, aes(x = highway, y = accident_count, fill = highway)) +
geom_bar(stat = "identity") +
labs(title = "Accident Count by highway Type", x = "Highway Type", y = "Number of Accidents") +
theme_minimal()
We will now single out the motorway to further determine what type of accidents are most common on them.
# Filter for motorway and count causes of accidents
motorway_accidents_cause_count <- accidents_on_roads %>%
filter(highway == "motorway") %>%
group_by(presumed_cause) %>%
summarise(count = n()) %>%
arrange(desc(count)) %>% # Sort by count in descending order
slice(1:5) # Select top 5 causes
# Create a bar plot for the top 5 causes of accidents on motorways
ggplot(motorway_accidents_cause_count, aes(x = reorder(presumed_cause, count), y = count, fill = presumed_cause)) +
geom_bar(stat = "identity") +
coord_flip() + # Flip the axes for better readability
labs(title = "Top 5 Causes of Accidents on Motorways",
x = "Accident Cause",
y = "Count") +
theme_minimal() +
theme(legend.position = "none") # Hide legend
Speeding seems to be by far the most common reason for road accidents across the region.
The best way to deal with this would be the installation of more speed cameras and harsher penalties, in the form of larger fines, for failing to adhere to regulations.
Currently, fines for speeding can be up to 4000B. Increasing this fine would be prudent and help reduce the amount of speeding accidents on motorways and across the region in general.
Interactive map to visualize Presumed Cause
To further reinforce our expectations, we will produce a map that allows us to hover over all regions and visualize the causes of accidents in each polygon.
We first perform a spatial join using the st_join function and then count the number of accidents based on the presumed_cause.
# Perform a spatial join to count the number of accidents within each grid cell
accidents_by_grid <- st_join(bmr_grid_sf, bmr_accsf) %>%
group_by(geometry) %>%
summarise(accident_count = n()) # Count accidents in each grid cell
# Perform a spatial join to associate accidents with polygons
# This assumes you already have polygons in `bmr_grid_sf` and accidents in `bmr_accsf`
accidents_by_polygon_and_cause <- st_join(bmr_grid_sf, bmr_accsf) %>%
group_by(geometry, presumed_cause) %>%
summarise(cause_count = n()) # Count the number of accidents by presumed cause in each polygon
# Optionally, aggregate presumed causes for each polygon into a summary
polygon_cause_summary <- accidents_by_polygon_and_cause %>%
group_by(geometry) %>%
summarise(total_accidents = sum(cause_count),
presumed_cause_summary = paste(presumed_cause, ":", cause_count, collapse = "; "))# Switch to interactive mode to enable tooltips
tmap_mode("plot")
# Create the choropleth map with hover tooltips
tm_shape(polygon_cause_summary) +
tm_polygons(col = "total_accidents", palette = "Reds", style = "quantile",
title = "Accident Count", border.col = "black",
id = "presumed_cause_summary") + # Tooltip will show presumed causes
tm_layout(frame = FALSE, legend.outside = TRUE)
tmap_mode('plot')Kernel Density Estimation (KDE)
KDE will allow us to better estimate the distribution of accidents acrosss the Bangkok Metropolitan Region. Using this, we can make more informed decisions on where to focus resources on to reduce the rate of accidents across the region.
We start off by preparing the data for KDE.
# Step 3: Define the window (bounding box) from the boundary
boundary_bbox <- bbox(boundary_sp) # Get bounding box of boundary
window <- owin(xrange = boundary_bbox[1, ], yrange = boundary_bbox[2, ]) # Define window
# Step 4: Extract the coordinates from the SpatialPoints object
coordinates <- coordinates(bmracc_sp)
# Step 5: Convert the SpatialPoints into a ppp object using the window
bmracc_ppp <- ppp(x = coordinates[,1], y = coordinates[,2], window = window)
# Step 6: Check the structure of the resulting point pattern object
summary(bmracc_ppp)Planar point pattern: 12986 points
Average intensity 1.101448e-06 points per square unit
*Pattern contains duplicated points*
Coordinates are given to 10 decimal places
Window: rectangle = [587893.5, 712440.5] x [1484413.7, 1579076.3] units
(124500 x 94660 units)
Window area = 11789900000 square units# You can now use the ppp object for spatial analysis with spatstatNote that you must ensure to convert the spatial objects into ppp (planar point pattern) form before proceeding with Kernel Density Estimation. The spatstat package requires data to be in this format before conducting point pattern analysis. This also improves computational efficiency.
After creating the above PPP object, we check for duplicates by implementing the code chunk below.
any(duplicated(bmracc_ppp))[1] TRUEThe multiplicity() function helps give us a better insight into the duplicates that are present in the ppp object.
multiplicity(bmracc_ppp) 1 2 3 4 5 6 7 8 9 10 11 12 13
1 1 1 2 1 2 1 1 2 1 1 1 1
14 15 16 17 18 19 20 21 22 23 24 25 26
1 1 1 1 1 2 1 1 1 1 1 1 1
27 28 29 30 31 32 33 34 35 36 37 38 39
1 1 5 1 1 1 4 1 1 6 1 5 5
40 41 42 43 44 45 46 47 48 49 50 51 52
1 1 5 1 1 1 1 1 1 1 1 1 1
53 54 55 56 57 58 59 60 61 62 63 64 65
1 1 1 1 6 6 1 6 1 1 6 3 1
66 67 68 69 70 71 72 73 74 75 76 77 78
6 1 1 1 1 1 1 1 3 2 1 5 3
79 80 81 82 83 84 85 86 87 88 89 90 91
1 1 1 1 1 1 1 1 1 1 1 1 3
92 93 94 95 96 97 98 99 100 101 102 103 104
1 1 1 3 1 3 2 1 3 2 1 6 1
105 106 107 108 109 110 111 112 113 114 115 116 117
1 1 4 1 1 1 11 11 4 1 1 1 1
118 119 120 121 122 123 124 125 126 127 128 129 130
4 1 1 2 11 11 2 1 1 1 1 1 2
131 132 133 134 135 136 137 138 139 140 141 142 143
1 1 1 1 11 6 1 11 1 1 11 1 1
144 145 146 147 148 149 150 151 152 153 154 155 156
11 11 2 3 1 1 1 1 1 1 1 1 11
157 158 159 160 161 162 163 164 165 166 167 168 169
4 7 1 1 1 1 1 7 3 1 1 1 1
170 171 172 173 174 175 176 177 178 179 180 181 182
1 14 3 4 3 4 1 3 1 1 1 4 4
183 184 185 186 187 188 189 190 191 192 193 194 195
1 1 3 1 4 1 1 2 1 1 4 4 1
196 197 198 199 200 201 202 203 204 205 206 207 208
1 3 2 1 14 1 3 1 1 4 4 1 1
209 210 211 212 213 214 215 216 217 218 219 220 221
4 1 1 1 1 1 1 1 1 1 1 12 2
222 223 224 225 226 227 228 229 230 231 232 233 234
1 1 1 2 4 3 1 1 1 1 1 1 1
235 236 237 238 239 240 241 242 243 244 245 246 247
1 1 1 1 1 1 1 1 2 1 1 1 1
248 249 250 251 252 253 254 255 256 257 258 259 260
9 1 1 1 1 1 1 1 2 1 1 1 1
261 262 263 264 265 266 267 268 269 270 271 272 273
1 1 1 1 1 1 1 2 1 1 1 1 1
274 275 276 277 278 279 280 281 282 283 284 285 286
1 4 1 1 1 1 1 1 1 1 1 1 1
287 288 289 290 291 292 293 294 295 296 297 298 299
1 1 1 1 1 1 1 1 1 1 1 1 1
300 301 302 303 304 305 306 307 308 309 310 311 312
1 1 1 2 1 1 1 1 1 1 1 1 1
313 314 315 316 317 318 319 320 321 322 323 324 325
2 1 1 1 1 1 1 2 1 1 2 3 1
326 327 328 329 330 331 332 333 334 335 336 337 338
1 1 1 1 2 1 1 1 1 1 1 1 1
339 340 341 342 343 344 345 346 347 348 349 350 351
1 1 1 2 1 1 1 1 1 1 1 1 1
352 353 354 355 356 357 358 359 360 361 362 363 364
6 1 1 3 1 1 1 1 1 2 1 1 1
365 366 367 368 369 370 371 372 373 374 375 376 377
1 1 3 2 1 1 1 1 1 1 6 1 1
378 379 380 381 382 383 384 385 386 387 388 389 390
1 1 1 1 1 5 2 1 1 1 1 1 1
391 392 393 394 395 396 397 398 399 400 401 402 403
1 12 2 1 2 1 1 1 2 1 1 2 1
404 405 406 407 408 409 410 411 412 413 414 415 416
1 1 1 2 1 1 1 2 1 1 2 1 1
417 418 419 420 421 422 423 424 425 426 427 428 429
1 2 3 1 2 1 1 1 1 1 1 1 2
430 431 432 433 434 435 436 437 438 439 440 441 442
1 1 1 2 1 1 1 1 1 1 1 2 1
443 444 445 446 447 448 449 450 451 452 453 454 455
1 1 1 1 1 1 1 1 1 1 1 1 1
456 457 458 459 460 461 462 463 464 465 466 467 468
1 1 1 1 1 1 1 1 1 1 1 1 1
469 470 471 472 473 474 475 476 477 478 479 480 481
1 3 1 1 1 1 1 1 1 1 1 1 1
482 483 484 485 486 487 488 489 490 491 492 493 494
1 1 1 1 2 1 2 1 1 2 1 1 1
495 496 497 498 499 500 501 502 503 504 505 506 507
1 1 1 1 1 1 1 1 1 1 1 1 3
508 509 510 511 512 513 514 515 516 517 518 519 520
1 1 1 1 1 1 1 1 1 1 1 1 1
521 522 523 524 525 526 527 528 529 530 531 532 533
1 1 1 3 1 1 1 1 1 1 1 1 1
534 535 536 537 538 539 540 541 542 543 544 545 546
1 1 1 2 1 1 1 1 1 1 1 1 1
547 548 549 550 551 552 553 554 555 556 557 558 559
1 1 1 1 1 2 1 1 3 1 1 1 1
560 561 562 563 564 565 566 567 568 569 570 571 572
1 1 1 1 1 1 3 1 1 1 1 1 2
573 574 575 576 577 578 579 580 581 582 583 584 585
1 1 1 1 1 1 1 1 1 2 1 1 9
586 587 588 589 590 591 592 593 594 595 596 597 598
1 2 1 1 1 1 1 1 1 1 1 1 2
599 600 601 602 603 604 605 606 607 608 609 610 611
1 1 2 1 1 1 1 1 1 1 1 1 2
612 613 614 615 616 617 618 619 620 621 622 623 624
1 1 2 1 1 1 1 1 1 1 1 1 2
625 626 627 628 629 630 631 632 633 634 635 636 637
1 1 1 1 1 1 1 1 1 1 1 1 1
638 639 640 641 642 643 644 645 646 647 648 649 650
1 1 12 1 1 6 1 1 1 1 2 1 1
651 652 653 654 655 656 657 658 659 660 661 662 663
1 2 1 1 1 6 3 1 1 1 1 1 1
664 665 666 667 668 669 670 671 672 673 674 675 676
1 14 2 1 1 1 1 1 1 1 1 1 1
677 678 679 680 681 682 683 684 685 686 687 688 689
1 1 1 1 1 1 1 2 1 1 4 1 1
690 691 692 693 694 695 696 697 698 699 700 701 702
1 1 1 1 1 1 1 1 1 3 1 1 1
703 704 705 706 707 708 709 710 711 712 713 714 715
1 3 1 1 1 1 1 1 1 1 1 1 1
716 717 718 719 720 721 722 723 724 725 726 727 728
1 1 1 1 1 1 2 1 4 1 1 2 1
729 730 731 732 733 734 735 736 737 738 739 740 741
1 1 1 2 2 1 1 1 1 4 1 1 1
742 743 744 745 746 747 748 749 750 751 752 753 754
1 1 1 1 1 2 1 1 1 1 1 1 1
755 756 757 758 759 760 761 762 763 764 765 766 767
8 1 1 4 1 3 1 1 1 1 1 1 1
768 769 770 771 772 773 774 775 776 777 778 779 780
6 1 1 1 1 2 1 1 1 1 1 1 1
781 782 783 784 785 786 787 788 789 790 791 792 793
1 1 4 1 1 1 1 2 6 1 1 1 1
794 795 796 797 798 799 800 801 802 803 804 805 806
1 1 1 1 1 4 1 1 1 1 1 1 1
807 808 809 810 811 812 813 814 815 816 817 818 819
1 2 1 1 1 2 1 1 1 1 1 1 1
820 821 822 823 824 825 826 827 828 829 830 831 832
1 1 1 1 1 1 1 1 1 2 1 1 1
833 834 835 836 837 838 839 840 841 842 843 844 845
1 1 1 1 1 3 1 1 2 1 1 1 1
846 847 848 849 850 851 852 853 854 855 856 857 858
1 1 1 1 1 6 1 1 1 1 1 1 1
859 860 861 862 863 864 865 866 867 868 869 870 871
1 1 1 3 2 1 1 1 1 1 1 2 1
872 873 874 875 876 877 878 879 880 881 882 883 884
1 1 1 1 1 1 1 1 1 1 3 1 3
885 886 887 888 889 890 891 892 893 894 895 896 897
1 1 3 1 1 1 1 1 1 1 1 1 1
898 899 900 901 902 903 904 905 906 907 908 909 910
1 10 1 1 2 1 1 2 1 1 1 1 1
911 912 913 914 915 916 917 918 919 920 921 922 923
2 2 1 1 1 1 1 1 1 1 1 1 3
924 925 926 927 928 929 930 931 932 933 934 935 936
1 1 1 1 1 1 1 1 3 1 1 1 1
937 938 939 940 941 942 943 944 945 946 947 948 949
1 1 1 1 6 1 1 1 1 1 1 1 1
950 951 952 953 954 955 956 957 958 959 960 961 962
1 1 1 1 1 1 1 1 1 1 1 1 1
963 964 965 966 967 968 969 970 971 972 973 974 975
2 1 2 1 3 1 1 1 1 1 1 1 1
976 977 978 979 980 981 982 983 984 985 986 987 988
1 1 1 1 1 3 2 1 1 1 1 1 1
989 990 991 992 993 994 995 996 997 998 999 1000 1001
1 1 1 1 1 3 1 1 1 1 1 1 1
1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014
1 1 1 1 1 3 1 1 1 1 1 1 1
1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027
1 1 1 1 1 1 1 1 1 1 1 1 1
1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040
1 1 1 1 2 1 1 1 1 1 1 2 1
1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053
1 1 1 1 1 1 1 1 2 1 1 1 1
1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066
1 1 1 1 2 1 1 1 1 1 1 1 1
1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079
1 3 1 1 1 1 1 1 1 1 1 1 1
1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092
1 1 3 1 1 1 1 2 1 3 1 2 1
1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105
8 1 2 1 2 2 1 1 1 1 4 1 1
1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118
2 1 1 1 1 1 1 1 1 1 1 1 1
1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131
1 1 1 1 2 1 1 1 1 1 1 1 2
1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144
3 2 2 5 1 1 2 6 1 1 1 1 1
1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157
1 1 1 3 2 1 1 1 1 1 1 1 3
1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170
2 1 1 1 1 1 1 1 1 2 1 1 1
1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183
12 1 1 3 1 1 1 1 1 2 2 2 1
1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196
1 1 1 1 4 1 1 1 1 1 1 2 1
1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209
1 1 1 1 1 1 1 1 1 1 1 1 1
1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222
1 1 1 1 1 1 1 1 4 8 1 1 1
1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235
14 3 1 1 1 1 1 1 1 1 1 1 1
1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248
1 1 1 1 1 1 2 6 1 2 1 2 1
1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261
1 1 1 1 1 1 1 1 2 1 1 1 3
1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274
2 1 10 1 1 1 1 1 1 1 1 1 1
1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287
1 3 2 1 1 1 1 1 1 5 3 2 1
1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300
1 1 1 1 1 1 1 1 1 1 2 1 1
1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313
1 1 1 1 1 1 1 1 1 1 2 1 1
1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326
1 1 1 1 1 1 1 1 1 2 3 1 1
1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339
1 1 1 4 1 1 1 1 1 1 4 1 3
1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352
2 1 1 1 1 1 1 1 1 14 1 1 1
1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365
1 1 1 1 1 3 2 2 1 1 1 1 1
1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378
1 3 1 1 1 1 2 1 1 1 1 1 1
1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391
1 1 1 1 1 1 1 1 1 1 1 1 1
1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404
1 1 3 1 1 1 1 1 1 3 1 1 1
1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417
1 1 2 1 1 1 1 1 1 1 1 1 4
1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430
1 1 1 1 1 1 1 1 1 1 1 1 1
1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443
1 1 1 1 1 2 1 1 1 1 1 4 1
1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456
1 2 1 1 1 2 2 1 1 1 1 1 1
1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469
2 1 1 3 1 1 1 1 1 1 1 1 1
1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482
1 1 1 1 1 3 1 1 1 1 1 1 4
1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495
1 1 1 1 3 1 1 1 2 1 1 1 1
1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508
1 1 1 1 1 1 1 1 1 1 1 1 1
1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521
2 1 2 1 1 1 1 1 1 1 1 1 1
1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534
1 1 2 12 1 1 1 1 1 3 1 1 1
1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547
2 1 1 1 1 1 1 1 1 1 7 2 1
1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560
1 1 1 1 1 1 1 1 1 1 1 1 3
1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573
1 1 1 1 1 1 1 1 1 2 1 7 1
1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586
1 1 1 2 3 1 2 1 1 1 1 1 1
1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599
1 1 1 1 1 3 9 1 3 3 1 1 1
1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612
1 1 1 2 1 3 1 1 2 1 1 1 1
1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625
1 2 4 1 1 1 1 1 1 2 1 1 1
1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638
1 1 2 1 1 1 1 1 1 1 1 1 1
1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651
1 3 1 1 1 1 1 1 1 1 1 1 1
1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664
1 1 1 1 1 1 2 1 1 1 1 1 1
1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677
1 2 1 1 1 1 1 1 1 1 3 2 1
1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690
1 1 1 1 1 1 1 1 1 1 1 1 2
1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703
1 5 1 1 1 2 2 1 3 1 1 5 2
1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716
1 1 2 1 1 1 1 1 3 1 2 1 1
1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729
1 1 1 1 2 1 1 1 1 1 1 1 2
1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742
1 1 2 1 1 1 1 2 1 1 1 1 1
1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755
1 6 1 2 1 1 1 1 1 1 1 6 1
1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768
1 1 1 1 1 1 1 1 1 7 1 1 1
1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781
2 6 1 3 1 1 1 1 1 1 1 1 1
1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794
1 4 1 2 1 1 2 2 1 1 1 1 1
1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807
1 1 2 1 1 1 1 1 1 1 1 1 1
1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820
1 1 1 1 1 1 1 1 1 1 1 1 1
1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833
1 1 1 1 1 1 1 1 1 1 1 1 1
1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846
1 1 1 2 1 1 1 1 1 1 1 1 1
1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859
1 1 1 1 1 5 1 1 1 1 1 1 1
1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872
1 3 1 1 1 1 1 1 3 1 1 1 1
1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885
1 1 1 1 1 1 2 1 1 1 1 1 1
1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898
2 2 1 1 1 1 1 1 1 2 3 1 1
1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911
1 2 1 1 1 1 2 1 1 1 1 1 2
1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924
1 1 1 1 1 1 1 3 1 1 1 6 1
1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937
1 1 1 1 1 1 1 1 1 1 1 2 1
1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950
1 1 1 1 3 1 1 1 1 4 1 1 4
1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963
1 1 1 1 1 1 1 1 1 1 1 1 1
1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976
1 2 1 1 1 1 1 1 1 3 1 1 1
1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
1 1 1 1 1 3 1 2 1 1 2 2 1
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
1 1 1 1 1 1 1 1 1 1 1 1 1
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
1 1 1 1 1 1 1 4 1 1 1 2 2
2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028
1 1 1 1 3 1 1 4 1 1 1 1 4
2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041
2 1 1 1 1 1 1 1 1 1 1 1 1
2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054
2 1 3 1 1 1 2 2 1 1 1 1 1
2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067
3 1 1 2 1 1 1 1 1 1 1 1 1
2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080
1 1 1 1 1 1 1 1 1 1 1 3 1
2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093
6 1 1 1 1 1 1 1 1 1 1 2 2
2094 2095 2096 2097 2098 2099 2100 2101 2102 2103 2104 2105 2106
1 1 1 1 1 2 1 1 1 1 1 1 1
2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119
2 4 1 2 1 1 2 5 1 2 10 1 1
2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132
6 1 1 1 1 2 1 1 1 1 1 2 1
2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 2143 2144 2145
1 3 1 1 1 1 1 1 1 1 1 1 1
2146 2147 2148 2149 2150 2151 2152 2153 2154 2155 2156 2157 2158
3 1 1 1 1 1 1 2 1 1 1 1 1
2159 2160 2161 2162 2163 2164 2165 2166 2167 2168 2169 2170 2171
2 1 1 1 1 2 1 3 2 1 1 4 1
2172 2173 2174 2175 2176 2177 2178 2179 2180 2181 2182 2183 2184
1 1 1 1 1 14 1 1 1 1 1 1 1
2185 2186 2187 2188 2189 2190 2191 2192 2193 2194 2195 2196 2197
1 1 1 2 1 1 1 2 1 1 1 1 1
2198 2199 2200 2201 2202 2203 2204 2205 2206 2207 2208 2209 2210
1 2 1 2 1 1 1 1 1 1 1 2 1
2211 2212 2213 2214 2215 2216 2217 2218 2219 2220 2221 2222 2223
1 1 1 1 1 1 1 1 1 1 1 1 1
2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236
1 1 1 1 1 1 1 9 1 1 1 1 1
2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249
1 1 1 1 1 1 1 1 2 1 1 1 1
2250 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262
1 1 1 1 1 1 1 1 1 1 4 3 1
2263 2264 2265 2266 2267 2268 2269 2270 2271 2272 2273 2274 2275
1 1 1 1 1 2 1 1 1 1 4 1 4
2276 2277 2278 2279 2280 2281 2282 2283 2284 2285 2286 2287 2288
3 1 12 1 1 1 1 1 1 1 1 1 1
2289 2290 2291 2292 2293 2294 2295 2296 2297 2298 2299 2300 2301
1 1 1 1 1 1 1 1 1 2 1 1 1
2302 2303 2304 2305 2306 2307 2308 2309 2310 2311 2312 2313 2314
3 6 1 2 1 1 1 1 1 1 4 1 1
2315 2316 2317 2318 2319 2320 2321 2322 2323 2324 2325 2326 2327
1 1 1 1 1 1 1 1 1 1 1 1 1
2328 2329 2330 2331 2332 2333 2334 2335 2336 2337 2338 2339 2340
12 1 1 1 1 1 1 1 1 1 3 1 1
2341 2342 2343 2344 2345 2346 2347 2348 2349 2350 2351 2352 2353
1 1 1 1 1 1 1 4 1 1 1 1 1
2354 2355 2356 2357 2358 2359 2360 2361 2362 2363 2364 2365 2366
1 1 1 1 1 1 1 2 1 1 1 2 3
2367 2368 2369 2370 2371 2372 2373 2374 2375 2376 2377 2378 2379
1 1 1 3 3 1 1 2 1 1 1 1 1
2380 2381 2382 2383 2384 2385 2386 2387 2388 2389 2390 2391 2392
1 1 1 1 3 1 2 1 1 1 1 2 1
2393 2394 2395 2396 2397 2398 2399 2400 2401 2402 2403 2404 2405
1 1 1 1 1 1 1 1 1 1 1 1 1
2406 2407 2408 2409 2410 2411 2412 2413 2414 2415 2416 2417 2418
10 1 1 1 1 1 1 1 1 12 1 1 1
2419 2420 2421 2422 2423 2424 2425 2426 2427 2428 2429 2430 2431
1 1 3 1 1 1 1 1 1 1 1 1 1
2432 2433 2434 2435 2436 2437 2438 2439 2440 2441 2442 2443 2444
1 2 1 1 1 1 2 1 2 1 1 1 1
2445 2446 2447 2448 2449 2450 2451 2452 2453 2454 2455 2456 2457
1 1 1 1 1 1 1 1 1 1 1 1 2
2458 2459 2460 2461 2462 2463 2464 2465 2466 2467 2468 2469 2470
1 1 1 2 1 2 1 3 1 1 1 1 1
2471 2472 2473 2474 2475 2476 2477 2478 2479 2480 2481 2482 2483
1 1 1 1 1 1 4 8 6 1 3 1 1
2484 2485 2486 2487 2488 2489 2490 2491 2492 2493 2494 2495 2496
1 1 1 2 1 1 1 1 1 1 1 1 1
2497 2498 2499 2500 2501 2502 2503 2504 2505 2506 2507 2508 2509
1 1 1 1 1 1 1 1 1 2 9 1 1
2510 2511 2512 2513 2514 2515 2516 2517 2518 2519 2520 2521 2522
1 1 1 12 1 1 1 2 1 1 1 1 1
2523 2524 2525 2526 2527 2528 2529 2530 2531 2532 2533 2534 2535
1 3 1 1 1 1 2 3 1 4 1 1 1
2536 2537 2538 2539 2540 2541 2542 2543 2544 2545 2546 2547 2548
2 1 1 1 1 1 1 1 1 1 1 1 1
2549 2550 2551 2552 2553 2554 2555 2556 2557 2558 2559 2560 2561
2 1 2 1 1 1 1 1 1 1 1 1 4
2562 2563 2564 2565 2566 2567 2568 2569 2570 2571 2572 2573 2574
1 1 2 1 1 2 6 1 1 1 1 10 1
2575 2576 2577 2578 2579 2580 2581 2582 2583 2584 2585 2586 2587
1 1 3 1 1 1 1 1 1 1 1 1 1
2588 2589 2590 2591 2592 2593 2594 2595 2596 2597 2598 2599 2600
1 1 1 4 1 1 1 1 1 1 1 1 1
2601 2602 2603 2604 2605 2606 2607 2608 2609 2610 2611 2612 2613
1 1 1 1 1 1 1 1 2 1 1 1 1
2614 2615 2616 2617 2618 2619 2620 2621 2622 2623 2624 2625 2626
1 1 1 1 1 1 2 1 1 1 1 1 1
2627 2628 2629 2630 2631 2632 2633 2634 2635 2636 2637 2638 2639
3 1 1 1 1 1 1 1 1 1 1 2 1
2640 2641 2642 2643 2644 2645 2646 2647 2648 2649 2650 2651 2652
1 1 1 1 1 10 10 1 2 1 1 1 1
2653 2654 2655 2656 2657 2658 2659 2660 2661 2662 2663 2664 2665
1 1 3 1 1 1 1 1 1 1 1 1 2
2666 2667 2668 2669 2670 2671 2672 2673 2674 2675 2676 2677 2678
1 2 8 2 1 1 1 1 1 1 1 1 1
2679 2680 2681 2682 2683 2684 2685 2686 2687 2688 2689 2690 2691
1 1 1 1 1 1 1 1 1 2 1 3 1
2692 2693 2694 2695 2696 2697 2698 2699 2700 2701 2702 2703 2704
1 1 8 8 1 1 1 1 2 1 2 1 1
2705 2706 2707 2708 2709 2710 2711 2712 2713 2714 2715 2716 2717
1 1 1 6 1 1 1 1 1 1 1 1 1
2718 2719 2720 2721 2722 2723 2724 2725 2726 2727 2728 2729 2730
3 1 1 3 1 1 1 1 1 2 1 1 1
2731 2732 2733 2734 2735 2736 2737 2738 2739 2740 2741 2742 2743
1 1 1 1 2 1 3 2 1 1 2 1 1
2744 2745 2746 2747 2748 2749 2750 2751 2752 2753 2754 2755 2756
1 1 1 1 1 4 1 1 1 1 8 1 1
2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769
1 1 1 1 1 1 1 1 6 6 1 1 1
2770 2771 2772 2773 2774 2775 2776 2777 2778 2779 2780 2781 2782
1 3 1 3 1 1 1 2 1 1 1 1 6
2783 2784 2785 2786 2787 2788 2789 2790 2791 2792 2793 2794 2795
1 1 1 1 1 1 1 1 1 1 5 1 2
2796 2797 2798 2799 2800 2801 2802 2803 2804 2805 2806 2807 2808
1 1 1 1 1 1 1 1 1 1 1 1 1
2809 2810 2811 2812 2813 2814 2815 2816 2817 2818 2819 2820 2821
1 1 1 1 1 1 2 1 1 1 14 1 1
2822 2823 2824 2825 2826 2827 2828 2829 2830 2831 2832 2833 2834
1 1 1 1 3 1 2 1 1 5 2 1 1
2835 2836 2837 2838 2839 2840 2841 2842 2843 2844 2845 2846 2847
1 1 1 1 1 1 1 1 1 1 1 2 1
2848 2849 2850 2851 2852 2853 2854 2855 2856 2857 2858 2859 2860
1 1 1 1 1 1 1 2 1 1 1 1 2
2861 2862 2863 2864 2865 2866 2867 2868 2869 2870 2871 2872 2873
1 1 1 1 1 1 1 1 1 1 1 1 2
2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 2886
1 1 1 1 1 12 1 1 1 1 1 1 1
2887 2888 2889 2890 2891 2892 2893 2894 2895 2896 2897 2898 2899
1 1 1 1 1 2 1 1 4 1 1 2 1
2900 2901 2902 2903 2904 2905 2906 2907 2908 2909 2910 2911 2912
1 1 1 1 1 1 1 1 1 1 1 1 1
2913 2914 2915 2916 2917 2918 2919 2920 2921 2922 2923 2924 2925
1 1 1 1 1 2 1 1 1 1 1 1 3
2926 2927 2928 2929 2930 2931 2932 2933 2934 2935 2936 2937 2938
1 1 1 1 1 2 1 1 1 1 3 1 1
2939 2940 2941 2942 2943 2944 2945 2946 2947 2948 2949 2950 2951
1 1 1 1 1 1 1 1 1 1 1 1 1
2952 2953 2954 2955 2956 2957 2958 2959 2960 2961 2962 2963 2964
1 1 1 1 1 1 1 1 1 1 1 1 1
2965 2966 2967 2968 2969 2970 2971 2972 2973 2974 2975 2976 2977
1 1 1 1 1 1 1 1 1 1 1 1 1
2978 2979 2980 2981 2982 2983 2984 2985 2986 2987 2988 2989 2990
1 1 1 1 1 3 1 1 1 1 1 2 1
2991 2992 2993 2994 2995 2996 2997 2998 2999 3000 3001 3002 3003
2 1 1 1 1 1 1 1 1 1 1 1 1
3004 3005 3006 3007 3008 3009 3010 3011 3012 3013 3014 3015 3016
1 1 1 1 1 1 2 1 1 1 1 3 1
3017 3018 3019 3020 3021 3022 3023 3024 3025 3026 3027 3028 3029
1 1 1 1 1 1 1 1 2 1 1 1 1
3030 3031 3032 3033 3034 3035 3036 3037 3038 3039 3040 3041 3042
2 1 1 1 1 1 1 2 1 1 1 1 4
3043 3044 3045 3046 3047 3048 3049 3050 3051 3052 3053 3054 3055
1 1 1 1 1 1 1 1 1 1 1 1 1
3056 3057 3058 3059 3060 3061 3062 3063 3064 3065 3066 3067 3068
1 1 1 1 1 1 1 1 3 2 1 2 1
3069 3070 3071 3072 3073 3074 3075 3076 3077 3078 3079 3080 3081
1 1 1 1 1 1 1 1 1 1 1 1 1
3082 3083 3084 3085 3086 3087 3088 3089 3090 3091 3092 3093 3094
1 2 1 1 1 1 4 1 1 1 1 1 1
3095 3096 3097 3098 3099 3100 3101 3102 3103 3104 3105 3106 3107
1 1 1 1 1 1 1 1 1 1 4 1 1
3108 3109 3110 3111 3112 3113 3114 3115 3116 3117 3118 3119 3120
1 1 1 1 1 1 1 1 1 1 1 1 2
3121 3122 3123 3124 3125 3126 3127 3128 3129 3130 3131 3132 3133
2 1 2 1 3 2 1 1 1 1 1 1 4
3134 3135 3136 3137 3138 3139 3140 3141 3142 3143 3144 3145 3146
1 2 1 1 1 1 1 1 1 1 1 1 1
3147 3148 3149 3150 3151 3152 3153 3154 3155 3156 3157 3158 3159
1 1 2 1 1 3 1 1 1 1 1 1 1
3160 3161 3162 3163 3164 3165 3166 3167 3168 3169 3170 3171 3172
1 7 1 1 8 1 1 1 1 7 1 1 1
3173 3174 3175 3176 3177 3178 3179 3180 3181 3182 3183 3184 3185
1 1 1 1 1 2 3 1 2 4 1 1 7
3186 3187 3188 3189 3190 3191 3192 3193 3194 3195 3196 3197 3198
1 1 5 1 1 1 1 1 1 1 1 1 1
3199 3200 3201 3202 3203 3204 3205 3206 3207 3208 3209 3210 3211
1 2 1 1 1 1 1 1 8 14 1 1 1
3212 3213 3214 3215 3216 3217 3218 3219 3220 3221 3222 3223 3224
1 1 1 1 6 1 1 1 1 9 2 6 1
3225 3226 3227 3228 3229 3230 3231 3232 3233 3234 3235 3236 3237
2 1 1 1 1 1 1 1 1 8 1 2 1
3238 3239 3240 3241 3242 3243 3244 3245 3246 3247 3248 3249 3250
7 1 1 1 1 1 8 1 1 1 1 1 1
3251 3252 3253 3254 3255 3256 3257 3258 3259 3260 3261 3262 3263
1 1 1 1 1 1 1 4 1 2 1 1 2
3264 3265 3266 3267 3268 3269 3270 3271 3272 3273 3274 3275 3276
1 1 1 1 1 1 1 1 1 1 4 1 1
3277 3278 3279 3280 3281 3282 3283 3284 3285 3286 3287 3288 3289
1 1 5 2 1 1 3 1 1 1 12 1 1
3290 3291 3292 3293 3294 3295 3296 3297 3298 3299 3300 3301 3302
1 1 1 1 1 1 12 14 1 4 1 1 1
3303 3304 3305 3306 3307 3308 3309 3310 3311 3312 3313 3314 3315
1 1 1 2 1 2 1 1 1 1 1 1 1
3316 3317 3318 3319 3320 3321 3322 3323 3324 3325 3326 3327 3328
2 2 1 1 1 1 1 1 1 1 1 1 1
3329 3330 3331 3332 3333 3334 3335 3336 3337 3338 3339 3340 3341
1 2 1 5 3 1 2 2 1 1 1 1 1
3342 3343 3344 3345 3346 3347 3348 3349 3350 3351 3352 3353 3354
2 3 1 1 1 3 1 4 2 1 1 1 1
3355 3356 3357 3358 3359 3360 3361 3362 3363 3364 3365 3366 3367
1 1 1 3 1 1 2 3 1 1 4 1 1
3368 3369 3370 3371 3372 3373 3374 3375 3376 3377 3378 3379 3380
1 1 1 1 1 1 10 1 1 1 1 1 1
3381 3382 3383 3384 3385 3386 3387 3388 3389 3390 3391 3392 3393
1 1 2 1 3 1 1 1 9 1 1 1 2
3394 3395 3396 3397 3398 3399 3400 3401 3402 3403 3404 3405 3406
1 1 1 1 3 1 1 2 1 4 1 15 1
3407 3408 3409 3410 3411 3412 3413 3414 3415 3416 3417 3418 3419
2 1 1 1 5 2 1 1 1 1 1 1 1
3420 3421 3422 3423 3424 3425 3426 3427 3428 3429 3430 3431 3432
1 2 1 1 1 1 1 1 1 4 2 1 1
3433 3434 3435 3436 3437 3438 3439 3440 3441 3442 3443 3444 3445
1 1 1 1 1 3 1 1 1 1 1 1 1
3446 3447 3448 3449 3450 3451 3452 3453 3454 3455 3456 3457 3458
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3459 3460 3461 3462 3463 3464 3465 3466 3467 3468 3469 3470 3471
1 2 1 1 1 1 1 1 1 1 1 1 1
3472 3473 3474 3475 3476 3477 3478 3479 3480 3481 3482 3483 3484
4 1 5 1 1 2 1 1 1 1 1 1 1
3485 3486 3487 3488 3489 3490 3491 3492 3493 3494 3495 3496 3497
1 1 1 1 1 6 4 1 1 1 1 1 1
3498 3499 3500 3501 3502 3503 3504 3505 3506 3507 3508 3509 3510
1 2 1 1 1 1 1 1 4 1 1 1 1
3511 3512 3513 3514 3515 3516 3517 3518 3519 3520 3521 3522 3523
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3524 3525 3526 3527 3528 3529 3530 3531 3532 3533 3534 3535 3536
1 1 3 1 1 1 1 1 1 1 1 1 5
3537 3538 3539 3540 3541 3542 3543 3544 3545 3546 3547 3548 3549
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3550 3551 3552 3553 3554 3555 3556 3557 3558 3559 3560 3561 3562
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3563 3564 3565 3566 3567 3568 3569 3570 3571 3572 3573 3574 3575
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3576 3577 3578 3579 3580 3581 3582 3583 3584 3585 3586 3587 3588
1 1 1 1 5 1 1 6 1 1 1 1 1
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1 1 1 1 1 1 1 1 1 1 10 1 1
3602 3603 3604 3605 3606 3607 3608 3609 3610 3611 3612 3613 3614
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3615 3616 3617 3618 3619 3620 3621 3622 3623 3624 3625 3626 3627
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1 1 1 1 1 1 1 1 1 1 1 1 1
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3680 3681 3682 3683 3684 3685 3686 3687 3688 3689 3690 3691 3692
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3693 3694 3695 3696 3697 3698 3699 3700 3701 3702 3703 3704 3705
2 1 1 1 1 1 7 1 1 1 1 1 1
3706 3707 3708 3709 3710 3711 3712 3713 3714 3715 3716 3717 3718
1 1 2 1 2 2 1 1 1 4 1 1 2
3719 3720 3721 3722 3723 3724 3725 3726 3727 3728 3729 3730 3731
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3732 3733 3734 3735 3736 3737 3738 3739 3740 3741 3742 3743 3744
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1 2 1 1 1 1 1 10 1 1 1 1 1
3758 3759 3760 3761 3762 3763 3764 3765 3766 3767 3768 3769 3770
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3823 3824 3825 3826 3827 3828 3829 3830 3831 3832 3833 3834 3835
1 1 1 1 3 1 1 1 1 1 1 1 1
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3888 3889 3890 3891 3892 3893 3894 3895 3896 3897 3898 3899 3900
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3901 3902 3903 3904 3905 3906 3907 3908 3909 3910 3911 3912 3913
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3914 3915 3916 3917 3918 3919 3920 3921 3922 3923 3924 3925 3926
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3927 3928 3929 3930 3931 3932 3933 3934 3935 3936 3937 3938 3939
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3940 3941 3942 3943 3944 3945 3946 3947 3948 3949 3950 3951 3952
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3953 3954 3955 3956 3957 3958 3959 3960 3961 3962 3963 3964 3965
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3966 3967 3968 3969 3970 3971 3972 3973 3974 3975 3976 3977 3978
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3979 3980 3981 3982 3983 3984 3985 3986 3987 3988 3989 3990 3991
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3992 3993 3994 3995 3996 3997 3998 3999 4000 4001 4002 4003 4004
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4044 4045 4046 4047 4048 4049 4050 4051 4052 4053 4054 4055 4056
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4070 4071 4072 4073 4074 4075 4076 4077 4078 4079 4080 4081 4082
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4096 4097 4098 4099 4100 4101 4102 4103 4104 4105 4106 4107 4108
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4187 4188 4189 4190 4191 4192 4193 4194 4195 4196 4197 4198 4199
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4200 4201 4202 4203 4204 4205 4206 4207 4208 4209 4210 4211 4212
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4213 4214 4215 4216 4217 4218 4219 4220 4221 4222 4223 4224 4225
1 2 1 2 1 1 1 1 3 1 1 1 1
4226 4227 4228 4229 4230 4231 4232 4233 4234 4235 4236 4237 4238
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4239 4240 4241 4242 4243 4244 4245 4246 4247 4248 4249 4250 4251
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4252 4253 4254 4255 4256 4257 4258 4259 4260 4261 4262 4263 4264
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4265 4266 4267 4268 4269 4270 4271 4272 4273 4274 4275 4276 4277
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4278 4279 4280 4281 4282 4283 4284 4285 4286 4287 4288 4289 4290
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4291 4292 4293 4294 4295 4296 4297 4298 4299 4300 4301 4302 4303
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4304 4305 4306 4307 4308 4309 4310 4311 4312 4313 4314 4315 4316
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4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329
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4330 4331 4332 4333 4334 4335 4336 4337 4338 4339 4340 4341 4342
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4343 4344 4345 4346 4347 4348 4349 4350 4351 4352 4353 4354 4355
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4356 4357 4358 4359 4360 4361 4362 4363 4364 4365 4366 4367 4368
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4421 4422 4423 4424 4425 4426 4427 4428 4429 4430 4431 4432 4433
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7840 7841 7842 7843 7844 7845 7846 7847 7848 7849 7850 7851 7852
4 1 1 1 1 1 1 1 3 1 1 1 1
7853 7854 7855 7856 7857 7858 7859 7860 7861 7862 7863 7864 7865
1 1 1 1 1 1 1 1 1 3 1 1 1
7866 7867 7868 7869 7870 7871 7872 7873 7874 7875 7876 7877 7878
1 1 2 1 1 5 1 1 1 1 1 1 1
7879 7880 7881 7882 7883 7884 7885 7886 7887 7888 7889 7890 7891
1 2 2 1 1 1 1 1 1 1 5 1 1
7892 7893 7894 7895 7896 7897 7898 7899 7900 7901 7902 7903 7904
4 1 14 1 1 2 1 1 2 1 1 1 1
7905 7906 7907 7908 7909 7910 7911 7912 7913 7914 7915 7916 7917
1 1 1 1 1 2 5 1 1 1 1 1 1
7918 7919 7920 7921 7922 7923 7924 7925 7926 7927 7928 7929 7930
4 1 2 1 1 1 1 1 1 1 1 2 1
7931 7932 7933 7934 7935 7936 7937 7938 7939 7940 7941 7942 7943
1 1 2 1 1 1 1 2 4 1 4 1 1
7944 7945 7946 7947 7948 7949 7950 7951 7952 7953 7954 7955 7956
1 1 1 1 1 1 1 1 1 1 1 1 1
7957 7958 7959 7960 7961 7962 7963 7964 7965 7966 7967 7968 7969
1 1 1 2 1 1 1 1 1 1 1 1 1
7970 7971 7972 7973 7974 7975 7976 7977 7978 7979 7980 7981 7982
1 1 1 2 1 1 1 1 1 1 1 1 1
7983 7984 7985 7986 7987 7988 7989 7990 7991 7992 7993 7994 7995
10 1 1 2 5 1 1 1 1 1 1 1 1
7996 7997 7998 7999 8000 8001 8002 8003 8004 8005 8006 8007 8008
1 1 1 1 1 2 1 1 1 1 1 1 1
8009 8010 8011 8012 8013 8014 8015 8016 8017 8018 8019 8020 8021
1 1 1 3 1 1 1 3 1 1 1 1 1
8022 8023 8024 8025 8026 8027 8028 8029 8030 8031 8032 8033 8034
1 1 1 1 2 3 1 1 1 1 1 1 1
8035 8036 8037 8038 8039 8040 8041 8042 8043 8044 8045 8046 8047
1 2 1 1 6 1 1 1 1 1 1 1 1
8048 8049 8050 8051 8052 8053 8054 8055 8056 8057 8058 8059 8060
1 1 1 1 1 1 1 1 1 1 1 1 1
8061 8062 8063 8064 8065 8066 8067 8068 8069 8070 8071 8072 8073
2 1 1 1 1 2 1 1 1 1 1 1 1
8074 8075 8076 8077 8078 8079 8080 8081 8082 8083 8084 8085 8086
1 1 1 1 1 1 1 2 1 1 1 1 1
8087 8088 8089 8090 8091 8092 8093 8094 8095 8096 8097 8098 8099
1 4 1 4 1 1 5 1 1 1 1 1 1
8100 8101 8102 8103 8104 8105 8106 8107 8108 8109 8110 8111 8112
1 1 5 1 5 1 1 4 1 1 1 1 1
8113 8114 8115 8116 8117 8118 8119 8120 8121 8122 8123 8124 8125
1 1 5 1 1 1 1 1 1 2 1 3 1
8126 8127 8128 8129 8130 8131 8132 8133 8134 8135 8136 8137 8138
1 1 1 1 1 1 1 1 2 1 1 1 1
8139 8140 8141 8142 8143 8144 8145 8146 8147 8148 8149 8150 8151
1 1 1 3 1 1 1 1 1 1 1 1 1
8152 8153 8154 8155 8156 8157 8158 8159 8160 8161 8162 8163 8164
1 1 1 1 1 1 1 1 1 1 3 1 1
8165 8166 8167 8168 8169 8170 8171 8172 8173 8174 8175 8176 8177
1 5 3 1 1 1 14 1 2 1 3 1 1
8178 8179 8180 8181 8182 8183 8184 8185 8186 8187 8188 8189 8190
1 1 1 1 1 1 1 41 1 1 1 15 3
8191 8192 8193 8194 8195 8196 8197 8198 8199 8200 8201 8202 8203
3 1 1 1 1 1 13 3 13 1 2 1 1
8204 8205 8206 8207 8208 8209 8210 8211 8212 8213 8214 8215 8216
1 1 1 1 1 1 1 1 1 1 5 1 1
8217 8218 8219 8220 8221 8222 8223 8224 8225 8226 8227 8228 8229
1 1 1 1 1 1 1 1 2 4 1 1 1
8230 8231 8232 8233 8234 8235 8236 8237 8238 8239 8240 8241 8242
1 1 1 1 1 1 1 2 1 1 1 1 1
8243 8244 8245 8246 8247 8248 8249 8250 8251 8252 8253 8254 8255
1 1 2 1 1 1 1 1 1 1 1 3 3
8256 8257 8258 8259 8260 8261 8262 8263 8264 8265 8266 8267 8268
1 1 1 1 1 1 2 1 3 1 2 1 1
8269 8270 8271 8272 8273 8274 8275 8276 8277 8278 8279 8280 8281
1 1 3 2 1 5 1 1 1 1 1 1 1
8282 8283 8284 8285 8286 8287 8288 8289 8290 8291 8292 8293 8294
1 1 1 1 1 1 1 1 1 1 1 2 1
8295 8296 8297 8298 8299 8300 8301 8302 8303 8304 8305 8306 8307
2 2 1 1 1 1 1 1 1 4 1 10 13
8308 8309 8310 8311 8312 8313 8314 8315 8316 8317 8318 8319 8320
1 1 1 1 1 1 1 1 1 1 1 1 5
8321 8322 8323 8324 8325 8326 8327 8328 8329 8330 8331 8332 8333
1 1 1 41 1 1 1 1 1 2 1 1 1
8334 8335 8336 8337 8338 8339 8340 8341 8342 8343 8344 8345 8346
1 1 1 1 1 1 9 1 1 1 1 1 1
8347 8348 8349 8350 8351 8352 8353 8354 8355 8356 8357 8358 8359
1 3 1 1 13 1 1 1 1 1 1 3 1
8360 8361 8362 8363 8364 8365 8366 8367 8368 8369 8370 8371 8372
1 2 1 1 1 1 1 1 1 1 1 1 1
8373 8374 8375 8376 8377 8378 8379 8380 8381 8382 8383 8384 8385
1 1 1 1 1 1 3 1 1 1 1 1 1
8386 8387 8388 8389 8390 8391 8392 8393 8394 8395 8396 8397 8398
1 2 1 1 1 1 1 1 1 1 1 2 1
8399 8400 8401 8402 8403 8404 8405 8406 8407 8408 8409 8410 8411
4 1 1 1 3 1 1 1 1 1 1 1 1
8412 8413 8414 8415 8416 8417 8418 8419 8420 8421 8422 8423 8424
1 1 3 1 1 1 2 1 1 1 3 2 1
8425 8426 8427 8428 8429 8430 8431 8432 8433 8434 8435 8436 8437
11 1 1 1 1 1 1 3 1 1 3 1 1
8438 8439 8440 8441 8442 8443 8444 8445 8446 8447 8448 8449 8450
1 1 1 1 1 13 1 4 1 1 2 1 1
8451 8452 8453 8454 8455 8456 8457 8458 8459 8460 8461 8462 8463
5 1 1 1 1 1 1 3 1 2 3 2 1
8464 8465 8466 8467 8468 8469 8470 8471 8472 8473 8474 8475 8476
2 3 1 1 2 1 1 2 1 1 3 14 1
8477 8478 8479 8480 8481 8482 8483 8484 8485 8486 8487 8488 8489
1 1 1 3 1 1 1 1 1 1 10 2 1
8490 8491 8492 8493 8494 8495 8496 8497 8498 8499 8500 8501 8502
1 1 1 1 1 1 5 6 1 1 1 1 1
8503 8504 8505 8506 8507 8508 8509 8510 8511 8512 8513 8514 8515
1 1 1 1 1 1 1 4 1 1 1 1 1
8516 8517 8518 8519 8520 8521 8522 8523 8524 8525 8526 8527 8528
1 1 1 1 1 1 1 1 1 1 1 1 1
8529 8530 8531 8532 8533 8534 8535 8536 8537 8538 8539 8540 8541
1 2 1 1 1 1 1 3 1 2 2 1 41
8542 8543 8544 8545 8546 8547 8548 8549 8550 8551 8552 8553 8554
3 3 1 1 1 2 1 41 1 2 7 1 1
8555 8556 8557 8558 8559 8560 8561 8562 8563 8564 8565 8566 8567
2 1 1 1 5 3 2 1 3 1 10 1 3
8568 8569 8570 8571 8572 8573 8574 8575 8576 8577 8578 8579 8580
1 1 1 1 1 1 14 2 1 1 1 1 1
8581 8582 8583 8584 8585 8586 8587 8588 8589 8590 8591 8592 8593
1 2 1 1 1 2 1 1 5 1 1 2 1
8594 8595 8596 8597 8598 8599 8600 8601 8602 8603 8604 8605 8606
1 1 1 1 1 2 1 1 1 3 1 1 1
8607 8608 8609 8610 8611 8612 8613 8614 8615 8616 8617 8618 8619
1 1 1 1 1 4 1 1 1 1 1 41 1
8620 8621 8622 8623 8624 8625 8626 8627 8628 8629 8630 8631 8632
1 1 1 4 1 1 1 1 1 1 1 1 3
8633 8634 8635 8636 8637 8638 8639 8640 8641 8642 8643 8644 8645
4 1 1 1 5 1 1 1 1 1 1 1 1
8646 8647 8648 8649 8650 8651 8652 8653 8654 8655 8656 8657 8658
1 1 1 1 1 3 1 1 1 1 1 1 1
8659 8660 8661 8662 8663 8664 8665 8666 8667 8668 8669 8670 8671
1 1 1 1 1 1 1 1 1 1 1 1 1
8672 8673 8674 8675 8676 8677 8678 8679 8680 8681 8682 8683 8684
1 4 1 1 1 1 3 1 1 1 1 13 1
8685 8686 8687 8688 8689 8690 8691 8692 8693 8694 8695 8696 8697
1 1 1 1 1 1 1 1 1 1 1 5 13
8698 8699 8700 8701 8702 8703 8704 8705 8706 8707 8708 8709 8710
2 1 9 2 1 3 1 1 1 1 1 2 1
8711 8712 8713 8714 8715 8716 8717 8718 8719 8720 8721 8722 8723
1 1 1 1 12 1 1 1 1 1 3 1 3
8724 8725 8726 8727 8728 8729 8730 8731 8732 8733 8734 8735 8736
1 1 1 1 4 1 1 1 1 1 1 4 1
8737 8738 8739 8740 8741 8742 8743 8744 8745 8746 8747 8748 8749
1 1 1 1 1 1 1 1 4 3 1 1 3
8750 8751 8752 8753 8754 8755 8756 8757 8758 8759 8760 8761 8762
2 1 1 2 1 1 1 1 5 1 1 1 1
8763 8764 8765 8766 8767 8768 8769 8770 8771 8772 8773 8774 8775
1 1 1 1 1 1 1 1 1 18 1 1 1
8776 8777 8778 8779 8780 8781 8782 8783 8784 8785 8786 8787 8788
1 1 3 3 1 1 1 1 1 7 1 1 1
8789 8790 8791 8792 8793 8794 8795 8796 8797 8798 8799 8800 8801
1 1 1 1 1 1 2 1 2 1 1 1 1
8802 8803 8804 8805 8806 8807 8808 8809 8810 8811 8812 8813 8814
1 3 11 1 1 1 2 1 2 1 1 1 1
8815 8816 8817 8818 8819 8820 8821 8822 8823 8824 8825 8826 8827
1 1 1 1 1 2 1 1 1 1 1 1 1
8828 8829 8830 8831 8832 8833 8834 8835 8836 8837 8838 8839 8840
1 2 1 1 4 1 1 1 1 1 1 1 1
8841 8842 8843 8844 8845 8846 8847 8848 8849 8850 8851 8852 8853
1 3 1 1 3 1 1 3 9 1 1 1 1
8854 8855 8856 8857 8858 8859 8860 8861 8862 8863 8864 8865 8866
1 1 3 2 1 1 1 1 1 3 1 1 1
8867 8868 8869 8870 8871 8872 8873 8874 8875 8876 8877 8878 8879
2 1 1 1 1 1 1 1 3 1 1 1 1
8880 8881 8882 8883 8884 8885 8886 8887 8888 8889 8890 8891 8892
5 2 1 1 1 1 1 1 1 1 1 1 1
8893 8894 8895 8896 8897 8898 8899 8900 8901 8902 8903 8904 8905
18 2 1 1 1 1 2 3 1 1 1 2 1
8906 8907 8908 8909 8910 8911 8912 8913 8914 8915 8916 8917 8918
1 1 1 1 1 1 1 1 1 1 1 1 1
8919 8920 8921 8922 8923 8924 8925 8926 8927 8928 8929 8930 8931
1 2 1 2 1 1 7 1 1 1 1 1 1
8932 8933 8934 8935 8936 8937 8938 8939 8940 8941 8942 8943 8944
1 1 1 1 1 1 1 1 3 1 2 1 1
8945 8946 8947 8948 8949 8950 8951 8952 8953 8954 8955 8956 8957
1 3 18 8 2 1 1 3 1 1 1 1 4
8958 8959 8960 8961 8962 8963 8964 8965 8966 8967 8968 8969 8970
1 1 1 1 1 1 1 1 1 3 1 1 1
8971 8972 8973 8974 8975 8976 8977 8978 8979 8980 8981 8982 8983
1 1 1 1 1 1 1 1 1 1 1 1 1
8984 8985 8986 8987 8988 8989 8990 8991 8992 8993 8994 8995 8996
1 1 2 2 1 2 5 1 1 1 3 1 1
8997 8998 8999 9000 9001 9002 9003 9004 9005 9006 9007 9008 9009
1 1 1 1 1 2 13 3 1 3 1 1 1
9010 9011 9012 9013 9014 9015 9016 9017 9018 9019 9020 9021 9022
1 1 1 1 41 8 1 10 1 1 1 1 5
9023 9024 9025 9026 9027 9028 9029 9030 9031 9032 9033 9034 9035
1 1 1 1 1 1 1 1 1 1 1 2 1
9036 9037 9038 9039 9040 9041 9042 9043 9044 9045 9046 9047 9048
1 1 1 1 1 1 1 1 1 1 1 1 1
9049 9050 9051 9052 9053 9054 9055 9056 9057 9058 9059 9060 9061
1 2 2 1 1 1 1 1 1 1 1 2 1
9062 9063 9064 9065 9066 9067 9068 9069 9070 9071 9072 9073 9074
1 18 2 1 1 1 1 1 1 11 1 1 2
9075 9076 9077 9078 9079 9080 9081 9082 9083 9084 9085 9086 9087
1 1 1 1 3 1 3 1 1 1 3 1 1
9088 9089 9090 9091 9092 9093 9094 9095 9096 9097 9098 9099 9100
1 1 1 1 1 1 1 1 41 8 1 1 1
9101 9102 9103 9104 9105 9106 9107 9108 9109 9110 9111 9112 9113
1 1 3 2 1 1 1 2 2 1 1 6 1
9114 9115 9116 9117 9118 9119 9120 9121 9122 9123 9124 9125 9126
1 1 1 1 1 1 1 1 1 1 1 1 1
9127 9128 9129 9130 9131 9132 9133 9134 9135 9136 9137 9138 9139
1 1 1 1 1 1 1 1 1 3 1 1 1
9140 9141 9142 9143 9144 9145 9146 9147 9148 9149 9150 9151 9152
1 1 1 1 1 1 3 1 1 1 1 4 2
9153 9154 9155 9156 9157 9158 9159 9160 9161 9162 9163 9164 9165
1 3 1 1 1 1 1 1 1 1 5 1 1
9166 9167 9168 9169 9170 9171 9172 9173 9174 9175 9176 9177 9178
1 1 2 1 2 1 1 1 1 1 2 1 1
9179 9180 9181 9182 9183 9184 9185 9186 9187 9188 9189 9190 9191
1 1 1 1 1 1 1 1 1 1 1 1 1
9192 9193 9194 9195 9196 9197 9198 9199 9200 9201 9202 9203 9204
1 1 1 1 1 2 41 1 1 1 1 2 1
9205 9206 9207 9208 9209 9210 9211 9212 9213 9214 9215 9216 9217
3 1 3 1 2 1 1 1 1 3 1 1 1
9218 9219 9220 9221 9222 9223 9224 9225 9226 9227 9228 9229 9230
2 2 3 1 1 1 1 1 2 1 1 1 1
9231 9232 9233 9234 9235 9236 9237 9238 9239 9240 9241 9242 9243
1 1 1 1 1 1 1 1 1 7 1 1 1
9244 9245 9246 9247 9248 9249 9250 9251 9252 9253 9254 9255 9256
1 1 1 1 1 1 1 1 2 1 1 5 1
9257 9258 9259 9260 9261 9262 9263 9264 9265 9266 9267 9268 9269
1 1 1 1 1 1 1 1 2 2 1 14 1
9270 9271 9272 9273 9274 9275 9276 9277 9278 9279 9280 9281 9282
4 7 7 1 1 18 2 1 1 2 1 1 1
9283 9284 9285 9286 9287 9288 9289 9290 9291 9292 9293 9294 9295
1 1 1 1 1 1 8 1 1 1 1 1 1
9296 9297 9298 9299 9300 9301 9302 9303 9304 9305 9306 9307 9308
4 2 1 1 1 1 1 1 1 1 1 1 6
9309 9310 9311 9312 9313 9314 9315 9316 9317 9318 9319 9320 9321
1 1 1 1 2 1 1 1 1 1 1 1 1
9322 9323 9324 9325 9326 9327 9328 9329 9330 9331 9332 9333 9334
1 1 1 1 1 1 1 1 1 1 1 1 1
9335 9336 9337 9338 9339 9340 9341 9342 9343 9344 9345 9346 9347
2 1 1 1 1 1 1 2 1 1 1 1 1
9348 9349 9350 9351 9352 9353 9354 9355 9356 9357 9358 9359 9360
1 1 1 1 1 1 1 1 1 1 1 1 1
9361 9362 9363 9364 9365 9366 9367 9368 9369 9370 9371 9372 9373
1 1 1 1 1 1 1 1 1 1 1 1 1
9374 9375 9376 9377 9378 9379 9380 9381 9382 9383 9384 9385 9386
1 1 1 1 1 2 1 1 2 1 1 1 1
9387 9388 9389 9390 9391 9392 9393 9394 9395 9396 9397 9398 9399
6 1 1 1 1 1 1 1 1 1 4 1 1
9400 9401 9402 9403 9404 9405 9406 9407 9408 9409 9410 9411 9412
2 1 1 1 1 1 1 1 1 1 1 1 1
9413 9414 9415 9416 9417 9418 9419 9420 9421 9422 9423 9424 9425
1 1 1 1 1 1 1 1 1 1 1 1 1
9426 9427 9428 9429 9430 9431 9432 9433 9434 9435 9436 9437 9438
1 1 1 1 1 1 1 1 1 1 1 1 1
9439 9440 9441 9442 9443 9444 9445 9446 9447 9448 9449 9450 9451
1 1 1 1 1 7 1 1 1 1 1 1 1
9452 9453 9454 9455 9456 9457 9458 9459 9460 9461 9462 9463 9464
1 1 2 1 1 3 1 1 1 1 1 1 1
9465 9466 9467 9468 9469 9470 9471 9472 9473 9474 9475 9476 9477
1 1 1 1 1 1 1 1 1 1 1 2 1
9478 9479 9480 9481 9482 9483 9484 9485 9486 9487 9488 9489 9490
1 1 7 1 1 1 4 7 1 1 1 2 6
9491 9492 9493 9494 9495 9496 9497 9498 9499 9500 9501 9502 9503
1 1 2 1 1 1 4 1 2 1 1 1 1
9504 9505 9506 9507 9508 9509 9510 9511 9512 9513 9514 9515 9516
2 1 1 1 1 9 1 1 1 3 1 1 1
9517 9518 9519 9520 9521 9522 9523 9524 9525 9526 9527 9528 9529
1 1 1 1 1 1 1 1 1 1 2 1 1
9530 9531 9532 9533 9534 9535 9536 9537 9538 9539 9540 9541 9542
1 1 2 1 6 1 1 1 1 1 1 1 1
9543 9544 9545 9546 9547 9548 9549 9550 9551 9552 9553 9554 9555
1 2 1 1 1 1 1 1 1 1 2 1 1
9556 9557 9558 9559 9560 9561 9562 9563 9564 9565 9566 9567 9568
1 1 4 3 4 1 1 1 1 1 1 1 3
9569 9570 9571 9572 9573 9574 9575 9576 9577 9578 9579 9580 9581
1 1 1 1 1 1 2 1 1 1 1 1 1
9582 9583 9584 9585 9586 9587 9588 9589 9590 9591 9592 9593 9594
1 1 6 2 3 3 1 1 1 1 2 1 1
9595 9596 9597 9598 9599 9600 9601 9602 9603 9604 9605 9606 9607
1 1 7 1 2 1 7 1 1 1 1 1 1
9608 9609 9610 9611 9612 9613 9614 9615 9616 9617 9618 9619 9620
1 1 1 2 1 1 1 1 1 1 1 2 1
9621 9622 9623 9624 9625 9626 9627 9628 9629 9630 9631 9632 9633
1 2 1 1 4 1 1 1 1 2 1 1 1
9634 9635 9636 9637 9638 9639 9640 9641 9642 9643 9644 9645 9646
2 1 1 1 1 1 1 1 1 1 1 3 6
9647 9648 9649 9650 9651 9652 9653 9654 9655 9656 9657 9658 9659
1 1 1 2 1 1 1 1 1 14 10 1 3
9660 9661 9662 9663 9664 9665 9666 9667 9668 9669 9670 9671 9672
2 1 6 1 1 1 1 1 1 1 1 4 5
9673 9674 9675 9676 9677 9678 9679 9680 9681 9682 9683 9684 9685
2 1 1 1 1 1 1 2 1 1 1 1 1
9686 9687 9688 9689 9690 9691 9692 9693 9694 9695 9696 9697 9698
1 1 1 1 1 1 1 1 1 1 1 1 1
9699 9700 9701 9702 9703 9704 9705 9706 9707 9708 9709 9710 9711
1 1 1 6 1 1 1 1 1 1 1 2 1
9712 9713 9714 9715 9716 9717 9718 9719 9720 9721 9722 9723 9724
1 1 1 2 1 1 1 2 1 1 4 1 7
9725 9726 9727 9728 9729 9730 9731 9732 9733 9734 9735 9736 9737
1 4 5 1 1 1 1 1 1 2 1 1 1
9738 9739 9740 9741 9742 9743 9744 9745 9746 9747 9748 9749 9750
1 1 1 1 1 7 1 2 1 2 2 1 1
9751 9752 9753 9754 9755 9756 9757 9758 9759 9760 9761 9762 9763
1 1 1 1 1 1 1 1 1 1 2 1 1
9764 9765 9766 9767 9768 9769 9770 9771 9772 9773 9774 9775 9776
1 7 1 1 1 1 4 1 1 1 1 7 3
9777 9778 9779 9780 9781 9782 9783 9784 9785 9786 9787 9788 9789
1 1 1 1 2 1 3 1 1 1 1 3 2
9790 9791 9792 9793 9794 9795 9796 9797 9798 9799 9800 9801 9802
7 1 1 5 1 1 1 1 3 1 1 2 1
9803 9804 9805 9806 9807 9808 9809 9810 9811 9812 9813 9814 9815
1 1 1 1 1 1 1 1 2 1 1 1 1
9816 9817 9818 9819 9820 9821 9822 9823 9824 9825 9826 9827 9828
1 1 1 1 1 1 1 2 1 1 1 1 1
9829 9830 9831 9832 9833 9834 9835 9836 9837 9838 9839 9840 9841
1 1 1 1 6 1 1 1 1 1 3 1 2
9842 9843 9844 9845 9846 9847 9848 9849 9850 9851 9852 9853 9854
1 1 1 5 1 4 1 1 1 1 3 1 1
9855 9856 9857 9858 9859 9860 9861 9862 9863 9864 9865 9866 9867
4 3 1 1 1 1 1 10 1 1 2 1 1
9868 9869 9870 9871 9872 9873 9874 9875 9876 9877 9878 9879 9880
1 1 1 1 1 1 1 1 5 1 1 4 1
9881 9882 9883 9884 9885 9886 9887 9888 9889 9890 9891 9892 9893
1 1 1 1 1 14 2 3 2 1 1 1 2
9894 9895 9896 9897 9898 9899 9900 9901 9902 9903 9904 9905 9906
3 1 6 2 1 2 1 1 1 1 3 1 1
9907 9908 9909 9910 9911 9912 9913 9914 9915 9916 9917 9918 9919
1 1 2 6 1 1 1 1 1 3 1 1 1
9920 9921 9922 9923 9924 9925 9926 9927 9928 9929 9930 9931 9932
1 1 1 14 1 1 1 1 5 1 1 2 4
9933 9934 9935 9936 9937 9938 9939 9940 9941 9942 9943 9944 9945
1 1 1 1 1 1 1 1 1 1 1 1 1
9946 9947 9948 9949 9950 9951 9952 9953 9954 9955 9956 9957 9958
1 1 1 2 1 1 1 1 1 1 1 1 1
9959 9960 9961 9962 9963 9964 9965 9966 9967 9968 9969 9970 9971
1 1 1 1 1 1 1 1 1 1 2 1 1
9972 9973 9974 9975 9976 9977 9978 9979 9980 9981 9982 9983 9984
15 1 11 2 1 1 8 2 1 1 1 4 1
9985 9986 9987 9988 9989 9990 9991 9992 9993 9994 9995 9996 9997
1 1 1 1 2 2 1 1 1 1 1 11 1
9998 9999 10000 10001 10002 10003 10004 10005 10006 10007 10008 10009 10010
1 1 5 1 2 1 1 1 1 12 2 1 4
10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023
1 1 1 1 4 1 1 1 4 1 1 1 1
10024 10025 10026 10027 10028 10029 10030 10031 10032 10033 10034 10035 10036
1 1 1 1 1 1 3 2 1 1 1 1 7
10037 10038 10039 10040 10041 10042 10043 10044 10045 10046 10047 10048 10049
1 5 1 2 1 1 8 1 1 1 1 1 1
10050 10051 10052 10053 10054 10055 10056 10057 10058 10059 10060 10061 10062
1 1 1 1 1 6 1 1 1 1 1 1 1
10063 10064 10065 10066 10067 10068 10069 10070 10071 10072 10073 10074 10075
1 1 1 1 1 1 1 1 1 1 8 1 2
10076 10077 10078 10079 10080 10081 10082 10083 10084 10085 10086 10087 10088
1 1 3 14 1 1 1 2 2 3 2 1 12
10089 10090 10091 10092 10093 10094 10095 10096 10097 10098 10099 10100 10101
1 1 1 1 1 1 1 11 1 1 2 1 1
10102 10103 10104 10105 10106 10107 10108 10109 10110 10111 10112 10113 10114
1 1 2 1 1 1 1 1 7 1 1 6 1
10115 10116 10117 10118 10119 10120 10121 10122 10123 10124 10125 10126 10127
1 1 1 1 1 1 1 1 1 1 1 1 4
10128 10129 10130 10131 10132 10133 10134 10135 10136 10137 10138 10139 10140
1 1 2 1 1 1 1 1 1 1 1 8 1
10141 10142 10143 10144 10145 10146 10147 10148 10149 10150 10151 10152 10153
1 1 1 1 1 2 1 1 1 1 14 1 9
10154 10155 10156 10157 10158 10159 10160 10161 10162 10163 10164 10165 10166
1 3 2 1 1 3 1 1 1 1 1 1 1
10167 10168 10169 10170 10171 10172 10173 10174 10175 10176 10177 10178 10179
1 2 1 2 1 1 1 1 1 2 1 1 2
10180 10181 10182 10183 10184 10185 10186 10187 10188 10189 10190 10191 10192
2 1 1 1 1 1 4 3 1 1 1 1 1
10193 10194 10195 10196 10197 10198 10199 10200 10201 10202 10203 10204 10205
1 1 14 1 1 41 2 1 1 1 1 1 1
10206 10207 10208 10209 10210 10211 10212 10213 10214 10215 10216 10217 10218
1 1 1 1 1 1 4 1 1 1 1 1 1
10219 10220 10221 10222 10223 10224 10225 10226 10227 10228 10229 10230 10231
1 1 1 1 1 1 1 1 1 1 1 1 2
10232 10233 10234 10235 10236 10237 10238 10239 10240 10241 10242 10243 10244
1 1 1 1 1 1 1 1 1 1 1 1 1
10245 10246 10247 10248 10249 10250 10251 10252 10253 10254 10255 10256 10257
5 1 1 1 1 1 2 1 1 4 1 1 2
10258 10259 10260 10261 10262 10263 10264 10265 10266 10267 10268 10269 10270
1 2 1 1 1 1 1 3 1 1 1 1 1
10271 10272 10273 10274 10275 10276 10277 10278 10279 10280 10281 10282 10283
1 1 2 1 1 1 2 1 1 1 1 1 1
10284 10285 10286 10287 10288 10289 10290 10291 10292 10293 10294 10295 10296
1 1 1 1 1 1 1 1 1 1 1 2 1
10297 10298 10299 10300 10301 10302 10303 10304 10305 10306 10307 10308 10309
1 2 1 1 1 2 7 1 1 1 1 1 1
10310 10311 10312 10313 10314 10315 10316 10317 10318 10319 10320 10321 10322
1 1 1 1 1 1 1 1 1 1 3 1 1
10323 10324 10325 10326 10327 10328 10329 10330 10331 10332 10333 10334 10335
1 1 1 1 1 1 1 1 1 1 1 1 1
10336 10337 10338 10339 10340 10341 10342 10343 10344 10345 10346 10347 10348
1 2 1 1 1 1 1 1 1 1 2 1 1
10349 10350 10351 10352 10353 10354 10355 10356 10357 10358 10359 10360 10361
1 1 2 1 1 1 2 3 1 1 1 1 1
10362 10363 10364 10365 10366 10367 10368 10369 10370 10371 10372 10373 10374
1 1 1 1 1 6 1 1 1 1 1 1 1
10375 10376 10377 10378 10379 10380 10381 10382 10383 10384 10385 10386 10387
1 1 1 1 2 1 4 1 1 1 4 1 1
10388 10389 10390 10391 10392 10393 10394 10395 10396 10397 10398 10399 10400
1 1 1 1 1 1 1 1 1 1 1 1 1
10401 10402 10403 10404 10405 10406 10407 10408 10409 10410 10411 10412 10413
1 1 1 1 1 6 1 2 1 1 2 1 4
10414 10415 10416 10417 10418 10419 10420 10421 10422 10423 10424 10425 10426
1 1 1 1 1 12 1 1 1 1 1 2 41
10427 10428 10429 10430 10431 10432 10433 10434 10435 10436 10437 10438 10439
1 1 1 2 2 1 2 1 1 1 1 1 1
10440 10441 10442 10443 10444 10445 10446 10447 10448 10449 10450 10451 10452
1 1 1 1 1 1 1 1 1 1 1 1 1
10453 10454 10455 10456 10457 10458 10459 10460 10461 10462 10463 10464 10465
1 1 1 2 1 1 1 1 1 1 1 1 18
10466 10467 10468 10469 10470 10471 10472 10473 10474 10475 10476 10477 10478
1 1 1 1 5 2 1 1 1 1 1 1 1
10479 10480 10481 10482 10483 10484 10485 10486 10487 10488 10489 10490 10491
2 2 1 2 1 1 1 4 1 2 1 1 1
10492 10493 10494 10495 10496 10497 10498 10499 10500 10501 10502 10503 10504
5 1 5 1 1 4 2 1 12 2 1 1 1
10505 10506 10507 10508 10509 10510 10511 10512 10513 10514 10515 10516 10517
1 1 1 1 1 1 1 1 1 1 1 1 1
10518 10519 10520 10521 10522 10523 10524 10525 10526 10527 10528 10529 10530
1 1 1 1 2 1 2 3 1 1 1 1 6
10531 10532 10533 10534 10535 10536 10537 10538 10539 10540 10541 10542 10543
1 1 1 1 1 5 1 1 1 1 1 1 1
10544 10545 10546 10547 10548 10549 10550 10551 10552 10553 10554 10555 10556
1 4 2 1 1 1 1 4 1 1 1 1 1
10557 10558 10559 10560 10561 10562 10563 10564 10565 10566 10567 10568 10569
1 7 1 3 1 1 1 1 7 4 1 1 8
10570 10571 10572 10573 10574 10575 10576 10577 10578 10579 10580 10581 10582
1 1 1 1 1 1 1 1 1 1 1 1 1
10583 10584 10585 10586 10587 10588 10589 10590 10591 10592 10593 10594 10595
7 1 1 1 2 7 4 1 1 1 18 1 1
10596 10597 10598 10599 10600 10601 10602 10603 10604 10605 10606 10607 10608
2 1 1 1 1 1 5 1 3 1 2 2 1
10609 10610 10611 10612 10613 10614 10615 10616 10617 10618 10619 10620 10621
1 1 1 1 4 1 1 3 1 2 1 3 1
10622 10623 10624 10625 10626 10627 10628 10629 10630 10631 10632 10633 10634
2 1 1 1 1 1 1 3 1 1 1 1 1
10635 10636 10637 10638 10639 10640 10641 10642 10643 10644 10645 10646 10647
1 1 1 1 1 1 7 1 1 1 1 7 1
10648 10649 10650 10651 10652 10653 10654 10655 10656 10657 10658 10659 10660
1 1 1 1 1 1 1 1 1 1 4 4 1
10661 10662 10663 10664 10665 10666 10667 10668 10669 10670 10671 10672 10673
1 2 1 1 2 1 1 1 5 1 1 3 2
10674 10675 10676 10677 10678 10679 10680 10681 10682 10683 10684 10685 10686
1 3 1 1 1 1 1 1 1 1 1 2 1
10687 10688 10689 10690 10691 10692 10693 10694 10695 10696 10697 10698 10699
1 1 1 1 2 2 41 1 1 1 1 1 1
10700 10701 10702 10703 10704 10705 10706 10707 10708 10709 10710 10711 10712
4 1 1 1 1 1 2 1 1 1 1 1 2
10713 10714 10715 10716 10717 10718 10719 10720 10721 10722 10723 10724 10725
1 3 1 1 1 1 1 2 1 1 1 1 1
10726 10727 10728 10729 10730 10731 10732 10733 10734 10735 10736 10737 10738
1 1 1 41 1 2 1 1 1 1 1 2 1
10739 10740 10741 10742 10743 10744 10745 10746 10747 10748 10749 10750 10751
1 8 1 9 1 1 1 6 15 1 1 10 1
10752 10753 10754 10755 10756 10757 10758 10759 10760 10761 10762 10763 10764
1 2 1 1 1 1 1 1 1 1 1 1 1
10765 10766 10767 10768 10769 10770 10771 10772 10773 10774 10775 10776 10777
1 1 1 1 1 3 18 3 1 41 4 1 1
10778 10779 10780 10781 10782 10783 10784 10785 10786 10787 10788 10789 10790
1 1 1 1 1 1 1 1 1 1 2 1 1
10791 10792 10793 10794 10795 10796 10797 10798 10799 10800 10801 10802 10803
1 1 1 5 1 1 1 1 2 1 1 1 1
10804 10805 10806 10807 10808 10809 10810 10811 10812 10813 10814 10815 10816
1 1 1 1 2 1 1 2 1 1 1 1 1
10817 10818 10819 10820 10821 10822 10823 10824 10825 10826 10827 10828 10829
1 1 1 1 1 1 1 1 3 1 1 1 2
10830 10831 10832 10833 10834 10835 10836 10837 10838 10839 10840 10841 10842
1 2 1 1 1 1 1 2 1 1 1 4 1
10843 10844 10845 10846 10847 10848 10849 10850 10851 10852 10853 10854 10855
2 1 1 41 1 1 1 1 1 1 1 1 1
10856 10857 10858 10859 10860 10861 10862 10863 10864 10865 10866 10867 10868
1 1 1 1 1 4 1 1 1 2 1 2 9
10869 10870 10871 10872 10873 10874 10875 10876 10877 10878 10879 10880 10881
1 1 1 1 1 1 1 7 1 1 1 1 1
10882 10883 10884 10885 10886 10887 10888 10889 10890 10891 10892 10893 10894
1 3 2 4 1 2 1 1 1 2 1 1 2
10895 10896 10897 10898 10899 10900 10901 10902 10903 10904 10905 10906 10907
1 1 1 1 1 1 1 1 1 1 1 1 1
10908 10909 10910 10911 10912 10913 10914 10915 10916 10917 10918 10919 10920
1 1 8 1 1 1 1 1 1 1 1 1 2
10921 10922 10923 10924 10925 10926 10927 10928 10929 10930 10931 10932 10933
1 1 2 1 1 1 1 1 1 4 1 1 1
10934 10935 10936 10937 10938 10939 10940 10941 10942 10943 10944 10945 10946
1 2 2 1 1 1 1 3 4 1 4 7 1
10947 10948 10949 10950 10951 10952 10953 10954 10955 10956 10957 10958 10959
1 3 2 1 1 2 1 1 1 3 41 1 1
10960 10961 10962 10963 10964 10965 10966 10967 10968 10969 10970 10971 10972
1 1 1 1 1 1 1 7 1 1 3 3 1
10973 10974 10975 10976 10977 10978 10979 10980 10981 10982 10983 10984 10985
1 1 7 1 1 6 1 1 5 1 1 4 1
10986 10987 10988 10989 10990 10991 10992 10993 10994 10995 10996 10997 10998
1 1 1 1 4 1 1 1 1 1 1 1 1
10999 11000 11001 11002 11003 11004 11005 11006 11007 11008 11009 11010 11011
1 1 1 1 1 1 1 1 1 1 1 1 1
11012 11013 11014 11015 11016 11017 11018 11019 11020 11021 11022 11023 11024
1 1 1 1 1 1 1 1 1 2 1 1 1
11025 11026 11027 11028 11029 11030 11031 11032 11033 11034 11035 11036 11037
1 1 3 2 1 3 1 7 1 1 1 1 1
11038 11039 11040 11041 11042 11043 11044 11045 11046 11047 11048 11049 11050
1 1 1 5 1 1 2 1 3 1 1 1 1
11051 11052 11053 11054 11055 11056 11057 11058 11059 11060 11061 11062 11063
1 1 1 1 1 1 4 8 1 1 1 1 2
11064 11065 11066 11067 11068 11069 11070 11071 11072 11073 11074 11075 11076
1 1 2 1 1 1 5 1 1 14 1 1 7
11077 11078 11079 11080 11081 11082 11083 11084 11085 11086 11087 11088 11089
1 1 1 1 1 1 1 1 1 1 2 1 1
11090 11091 11092 11093 11094 11095 11096 11097 11098 11099 11100 11101 11102
1 1 4 1 1 1 1 1 1 1 14 1 8
11103 11104 11105 11106 11107 11108 11109 11110 11111 11112 11113 11114 11115
1 1 1 1 1 1 3 1 6 1 2 1 1
11116 11117 11118 11119 11120 11121 11122 11123 11124 11125 11126 11127 11128
1 1 1 1 2 2 2 1 1 1 1 3 1
11129 11130 11131 11132 11133 11134 11135 11136 11137 11138 11139 11140 11141
1 6 1 1 1 1 2 1 2 2 2 1 1
11142 11143 11144 11145 11146 11147 11148 11149 11150 11151 11152 11153 11154
1 1 1 4 2 1 1 1 1 1 1 1 2
11155 11156 11157 11158 11159 11160 11161 11162 11163 11164 11165 11166 11167
1 1 1 1 1 1 1 1 1 1 1 1 1
11168 11169 11170 11171 11172 11173 11174 11175 11176 11177 11178 11179 11180
1 1 2 1 1 1 1 1 1 3 2 1 1
11181 11182 11183 11184 11185 11186 11187 11188 11189 11190 11191 11192 11193
1 2 1 1 10 1 3 1 1 1 1 1 1
11194 11195 11196 11197 11198 11199 11200 11201 11202 11203 11204 11205 11206
1 1 2 1 1 3 1 1 1 1 1 1 1
11207 11208 11209 11210 11211 11212 11213 11214 11215 11216 11217 11218 11219
1 1 1 1 1 1 1 1 4 1 1 1 1
11220 11221 11222 11223 11224 11225 11226 11227 11228 11229 11230 11231 11232
1 2 2 1 1 2 1 1 1 1 1 1 3
11233 11234 11235 11236 11237 11238 11239 11240 11241 11242 11243 11244 11245
1 1 1 1 4 1 1 10 1 5 1 1 7
11246 11247 11248 11249 11250 11251 11252 11253 11254 11255 11256 11257 11258
1 2 2 1 1 1 3 1 1 1 1 1 2
11259 11260 11261 11262 11263 11264 11265 11266 11267 11268 11269 11270 11271
1 1 1 2 1 1 1 1 1 1 1 1 2
11272 11273 11274 11275 11276 11277 11278 11279 11280 11281 11282 11283 11284
1 2 1 1 1 1 2 5 1 1 1 1 1
11285 11286 11287 11288 11289 11290 11291 11292 11293 11294 11295 11296 11297
1 1 1 1 1 1 1 1 1 1 1 1 1
11298 11299 11300 11301 11302 11303 11304 11305 11306 11307 11308 11309 11310
1 1 1 2 1 1 1 1 1 1 7 1 1
11311 11312 11313 11314 11315 11316 11317 11318 11319 11320 11321 11322 11323
1 1 1 1 1 2 1 1 2 5 1 1 1
11324 11325 11326 11327 11328 11329 11330 11331 11332 11333 11334 11335 11336
1 41 1 1 1 1 1 1 1 1 15 1 1
11337 11338 11339 11340 11341 11342 11343 11344 11345 11346 11347 11348 11349
2 1 1 1 1 1 1 1 1 1 1 2 2
11350 11351 11352 11353 11354 11355 11356 11357 11358 11359 11360 11361 11362
1 1 12 1 1 1 1 1 5 1 1 1 1
11363 11364 11365 11366 11367 11368 11369 11370 11371 11372 11373 11374 11375
1 1 1 1 3 3 41 1 1 1 3 5 1
11376 11377 11378 11379 11380 11381 11382 11383 11384 11385 11386 11387 11388
1 1 1 1 1 1 2 1 1 1 1 1 1
11389 11390 11391 11392 11393 11394 11395 11396 11397 11398 11399 11400 11401
2 1 1 1 1 1 1 1 1 1 1 1 1
11402 11403 11404 11405 11406 11407 11408 11409 11410 11411 11412 11413 11414
1 1 4 1 1 1 1 1 1 1 1 3 3
11415 11416 11417 11418 11419 11420 11421 11422 11423 11424 11425 11426 11427
4 1 1 1 2 1 1 1 1 1 1 1 2
11428 11429 11430 11431 11432 11433 11434 11435 11436 11437 11438 11439 11440
1 1 1 1 1 41 1 1 1 5 1 3 1
11441 11442 11443 11444 11445 11446 11447 11448 11449 11450 11451 11452 11453
2 1 1 1 1 1 1 1 1 1 1 1 7
11454 11455 11456 11457 11458 11459 11460 11461 11462 11463 11464 11465 11466
7 1 1 1 1 2 1 1 1 2 1 1 1
11467 11468 11469 11470 11471 11472 11473 11474 11475 11476 11477 11478 11479
14 2 7 1 1 1 2 1 1 1 1 3 1
11480 11481 11482 11483 11484 11485 11486 11487 11488 11489 11490 11491 11492
2 2 1 1 1 3 1 1 1 1 1 1 1
11493 11494 11495 11496 11497 11498 11499 11500 11501 11502 11503 11504 11505
1 1 1 1 1 1 1 3 1 1 1 1 1
11506 11507 11508 11509 11510 11511 11512 11513 11514 11515 11516 11517 11518
1 1 1 1 1 1 1 2 1 1 1 1 1
11519 11520 11521 11522 11523 11524 11525 11526 11527 11528 11529 11530 11531
1 1 3 18 5 2 1 1 1 1 1 1 1
11532 11533 11534 11535 11536 11537 11538 11539 11540 11541 11542 11543 11544
1 1 1 1 1 1 2 3 1 1 1 1 1
11545 11546 11547 11548 11549 11550 11551 11552 11553 11554 11555 11556 11557
1 1 1 1 1 5 1 2 1 2 1 1 1
11558 11559 11560 11561 11562 11563 11564 11565 11566 11567 11568 11569 11570
2 1 1 5 1 3 2 1 2 1 1 1 5
11571 11572 11573 11574 11575 11576 11577 11578 11579 11580 11581 11582 11583
2 1 1 1 1 1 2 1 1 2 7 3 1
11584 11585 11586 11587 11588 11589 11590 11591 11592 11593 11594 11595 11596
2 1 1 1 2 5 3 3 1 1 2 1 2
11597 11598 11599 11600 11601 11602 11603 11604 11605 11606 11607 11608 11609
8 12 1 1 1 2 1 1 4 1 1 1 1
11610 11611 11612 11613 11614 11615 11616 11617 11618 11619 11620 11621 11622
4 1 1 1 1 1 1 1 1 1 1 3 1
11623 11624 11625 11626 11627 11628 11629 11630 11631 11632 11633 11634 11635
1 4 1 2 1 1 1 1 1 1 4 1 1
11636 11637 11638 11639 11640 11641 11642 11643 11644 11645 11646 11647 11648
1 1 1 1 11 1 3 1 3 4 1 1 4
11649 11650 11651 11652 11653 11654 11655 11656 11657 11658 11659 11660 11661
1 1 1 1 1 1 10 1 1 1 1 1 1
11662 11663 11664 11665 11666 11667 11668 11669 11670 11671 11672 11673 11674
1 1 1 7 7 1 1 3 1 1 2 1 1
11675 11676 11677 11678 11679 11680 11681 11682 11683 11684 11685 11686 11687
1 1 1 1 1 1 1 1 1 3 1 1 1
11688 11689 11690 11691 11692 11693 11694 11695 11696 11697 11698 11699 11700
1 1 1 1 1 1 1 1 1 1 1 1 1
11701 11702 11703 11704 11705 11706 11707 11708 11709 11710 11711 11712 11713
1 1 1 1 3 1 1 1 1 12 3 1 1
11714 11715 11716 11717 11718 11719 11720 11721 11722 11723 11724 11725 11726
9 1 1 1 2 1 1 1 2 1 1 1 1
11727 11728 11729 11730 11731 11732 11733 11734 11735 11736 11737 11738 11739
1 1 1 1 1 1 1 1 1 1 1 1 1
11740 11741 11742 11743 11744 11745 11746 11747 11748 11749 11750 11751 11752
1 1 1 1 1 1 1 1 1 11 1 1 1
11753 11754 11755 11756 11757 11758 11759 11760 11761 11762 11763 11764 11765
1 1 3 1 1 1 2 18 1 1 1 1 1
11766 11767 11768 11769 11770 11771 11772 11773 11774 11775 11776 11777 11778
1 1 1 1 1 1 4 2 2 1 5 1 1
11779 11780 11781 11782 11783 11784 11785 11786 11787 11788 11789 11790 11791
1 4 1 1 5 1 1 1 1 1 1 2 1
11792 11793 11794 11795 11796 11797 11798 11799 11800 11801 11802 11803 11804
1 1 1 2 1 1 1 1 1 1 1 2 1
11805 11806 11807 11808 11809 11810 11811 11812 11813 11814 11815 11816 11817
1 4 4 2 4 1 1 1 1 1 1 1 1
11818 11819 11820 11821 11822 11823 11824 11825 11826 11827 11828 11829 11830
1 1 1 8 1 1 2 1 1 1 1 1 1
11831 11832 11833 11834 11835 11836 11837 11838 11839 11840 11841 11842 11843
1 4 1 1 1 1 1 1 1 3 2 1 3
11844 11845 11846 11847 11848 11849 11850 11851 11852 11853 11854 11855 11856
2 1 4 1 3 1 1 1 1 1 3 1 2
11857 11858 11859 11860 11861 11862 11863 11864 11865 11866 11867 11868 11869
1 1 1 1 1 1 1 1 2 1 1 1 8
11870 11871 11872 11873 11874 11875 11876 11877 11878 11879 11880 11881 11882
1 4 3 1 1 1 1 1 2 1 1 1 10
11883 11884 11885 11886 11887 11888 11889 11890 11891 11892 11893 11894 11895
1 4 1 5 1 1 1 1 8 3 3 1 1
11896 11897 11898 11899 11900 11901 11902 11903 11904 11905 11906 11907 11908
1 1 1 1 1 1 1 1 2 1 1 1 2
11909 11910 11911 11912 11913 11914 11915 11916 11917 11918 11919 11920 11921
1 1 1 1 1 1 1 1 1 1 1 1 1
11922 11923 11924 11925 11926 11927 11928 11929 11930 11931 11932 11933 11934
1 1 2 1 1 1 4 1 1 1 5 3 1
11935 11936 11937 11938 11939 11940 11941 11942 11943 11944 11945 11946 11947
1 1 1 1 1 1 2 1 1 1 1 3 1
11948 11949 11950 11951 11952 11953 11954 11955 11956 11957 11958 11959 11960
1 7 1 1 1 2 1 1 1 1 1 1 1
11961 11962 11963 11964 11965 11966 11967 11968 11969 11970 11971 11972 11973
1 1 2 1 1 1 1 1 1 1 1 1 1
11974 11975 11976 11977 11978 11979 11980 11981 11982 11983 11984 11985 11986
1 1 1 1 1 2 2 1 1 1 1 1 41
11987 11988 11989 11990 11991 11992 11993 11994 11995 11996 11997 11998 11999
1 1 1 1 1 1 1 2 7 1 1 1 3
12000 12001 12002 12003 12004 12005 12006 12007 12008 12009 12010 12011 12012
2 4 12 1 15 1 1 3 1 1 1 1 1
12013 12014 12015 12016 12017 12018 12019 12020 12021 12022 12023 12024 12025
1 2 1 1 1 2 1 1 1 1 2 1 1
12026 12027 12028 12029 12030 12031 12032 12033 12034 12035 12036 12037 12038
1 14 1 1 1 1 1 1 1 1 1 1 1
12039 12040 12041 12042 12043 12044 12045 12046 12047 12048 12049 12050 12051
2 1 1 4 1 1 2 1 1 1 1 1 1
12052 12053 12054 12055 12056 12057 12058 12059 12060 12061 12062 12063 12064
1 1 1 1 41 1 1 2 1 1 1 2 1
12065 12066 12067 12068 12069 12070 12071 12072 12073 12074 12075 12076 12077
1 1 1 1 1 1 18 5 1 1 1 1 1
12078 12079 12080 12081 12082 12083 12084 12085 12086 12087 12088 12089 12090
3 1 2 1 3 3 1 1 2 1 1 1 1
12091 12092 12093 12094 12095 12096 12097 12098 12099 12100 12101 12102 12103
1 2 1 1 1 1 2 1 1 14 7 3 1
12104 12105 12106 12107 12108 12109 12110 12111 12112 12113 12114 12115 12116
1 1 1 1 1 1 1 1 10 2 1 1 1
12117 12118 12119 12120 12121 12122 12123 12124 12125 12126 12127 12128 12129
1 1 1 2 1 1 1 1 2 41 1 1 1
12130 12131 12132 12133 12134 12135 12136 12137 12138 12139 12140 12141 12142
1 7 1 1 1 1 1 1 1 1 12 7 1
12143 12144 12145 12146 12147 12148 12149 12150 12151 12152 12153 12154 12155
1 1 1 1 2 41 1 1 1 1 1 1 1
12156 12157 12158 12159 12160 12161 12162 12163 12164 12165 12166 12167 12168
1 1 1 1 1 1 1 1 41 1 1 1 2
12169 12170 12171 12172 12173 12174 12175 12176 12177 12178 12179 12180 12181
1 1 1 1 1 1 1 1 1 1 1 3 1
12182 12183 12184 12185 12186 12187 12188 12189 12190 12191 12192 12193 12194
1 1 2 1 1 1 1 3 1 2 1 1 1
12195 12196 12197 12198 12199 12200 12201 12202 12203 12204 12205 12206 12207
1 1 1 1 1 1 41 1 5 18 1 2 1
12208 12209 12210 12211 12212 12213 12214 12215 12216 12217 12218 12219 12220
1 2 1 1 1 1 1 1 1 1 1 1 1
12221 12222 12223 12224 12225 12226 12227 12228 12229 12230 12231 12232 12233
1 1 1 1 1 1 3 2 1 2 1 1 1
12234 12235 12236 12237 12238 12239 12240 12241 12242 12243 12244 12245 12246
3 1 1 2 1 1 1 1 1 1 1 1 2
12247 12248 12249 12250 12251 12252 12253 12254 12255 12256 12257 12258 12259
1 1 2 1 1 1 1 1 5 1 1 1 1
12260 12261 12262 12263 12264 12265 12266 12267 12268 12269 12270 12271 12272
1 1 2 5 1 1 1 1 1 1 1 1 1
12273 12274 12275 12276 12277 12278 12279 12280 12281 12282 12283 12284 12285
5 1 2 5 2 1 1 1 1 3 1 1 1
12286 12287 12288 12289 12290 12291 12292 12293 12294 12295 12296 12297 12298
1 1 1 1 1 1 1 3 1 1 1 1 1
12299 12300 12301 12302 12303 12304 12305 12306 12307 12308 12309 12310 12311
1 1 1 1 1 6 4 1 1 1 1 1 1
12312 12313 12314 12315 12316 12317 12318 12319 12320 12321 12322 12323 12324
1 1 1 2 1 1 1 1 1 1 2 2 1
12325 12326 12327 12328 12329 12330 12331 12332 12333 12334 12335 12336 12337
1 1 1 2 1 1 1 1 1 1 1 2 1
12338 12339 12340 12341 12342 12343 12344 12345 12346 12347 12348 12349 12350
1 1 1 1 1 1 1 1 1 1 1 1 1
12351 12352 12353 12354 12355 12356 12357 12358 12359 12360 12361 12362 12363
1 3 3 1 1 1 2 1 1 2 1 1 1
12364 12365 12366 12367 12368 12369 12370 12371 12372 12373 12374 12375 12376
1 2 1 1 1 1 1 1 1 1 1 1 1
12377 12378 12379 12380 12381 12382 12383 12384 12385 12386 12387 12388 12389
1 1 1 1 1 1 1 1 1 1 1 1 1
12390 12391 12392 12393 12394 12395 12396 12397 12398 12399 12400 12401 12402
1 1 1 1 1 1 2 2 1 1 2 1 2
12403 12404 12405 12406 12407 12408 12409 12410 12411 12412 12413 12414 12415
1 1 1 1 2 1 1 1 1 1 1 1 3
12416 12417 12418 12419 12420 12421 12422 12423 12424 12425 12426 12427 12428
1 1 1 1 1 2 4 1 1 2 2 1 1
12429 12430 12431 12432 12433 12434 12435 12436 12437 12438 12439 12440 12441
1 1 1 1 1 1 2 1 1 1 1 1 1
12442 12443 12444 12445 12446 12447 12448 12449 12450 12451 12452 12453 12454
1 1 1 1 1 1 1 1 1 1 1 1 1
12455 12456 12457 12458 12459 12460 12461 12462 12463 12464 12465 12466 12467
1 1 1 1 1 1 1 1 1 1 1 1 1
12468 12469 12470 12471 12472 12473 12474 12475 12476 12477 12478 12479 12480
2 1 1 1 1 1 1 1 1 1 1 1 1
12481 12482 12483 12484 12485 12486 12487 12488 12489 12490 12491 12492 12493
1 2 1 1 1 1 2 1 1 1 1 1 3
12494 12495 12496 12497 12498 12499 12500 12501 12502 12503 12504 12505 12506
1 1 2 1 3 1 2 1 1 1 1 1 1
12507 12508 12509 12510 12511 12512 12513 12514 12515 12516 12517 12518 12519
1 1 1 3 2 1 1 1 1 3 1 1 1
12520 12521 12522 12523 12524 12525 12526 12527 12528 12529 12530 12531 12532
1 1 1 1 1 1 1 1 1 1 1 1 3
12533 12534 12535 12536 12537 12538 12539 12540 12541 12542 12543 12544 12545
1 1 1 2 1 1 4 1 1 1 2 1 1
12546 12547 12548 12549 12550 12551 12552 12553 12554 12555 12556 12557 12558
1 1 1 1 1 1 1 1 1 1 1 2 1
12559 12560 12561 12562 12563 12564 12565 12566 12567 12568 12569 12570 12571
1 3 1 1 1 2 1 2 1 1 1 1 1
12572 12573 12574 12575 12576 12577 12578 12579 12580 12581 12582 12583 12584
1 2 1 1 1 1 2 2 3 1 1 1 2
12585 12586 12587 12588 12589 12590 12591 12592 12593 12594 12595 12596 12597
1 1 1 1 1 1 1 2 1 2 2 1 1
12598 12599 12600 12601 12602 12603 12604 12605 12606 12607 12608 12609 12610
1 1 1 2 1 1 1 1 1 1 5 1 1
12611 12612 12613 12614 12615 12616 12617 12618 12619 12620 12621 12622 12623
1 1 1 1 6 1 1 1 1 1 1 2 1
12624 12625 12626 12627 12628 12629 12630 12631 12632 12633 12634 12635 12636
2 1 1 1 1 1 1 1 1 1 1 1 1
12637 12638 12639 12640 12641 12642 12643 12644 12645 12646 12647 12648 12649
1 1 6 1 1 2 1 1 1 1 1 1 5
12650 12651 12652 12653 12654 12655 12656 12657 12658 12659 12660 12661 12662
1 1 1 2 1 2 1 1 1 1 1 1 1
12663 12664 12665 12666 12667 12668 12669 12670 12671 12672 12673 12674 12675
1 1 1 1 2 1 1 1 1 1 1 3 1
12676 12677 12678 12679 12680 12681 12682 12683 12684 12685 12686 12687 12688
1 1 2 1 3 1 2 7 2 1 1 2 1
12689 12690 12691 12692 12693 12694 12695 12696 12697 12698 12699 12700 12701
1 4 7 7 1 1 1 1 1 1 1 1 1
12702 12703 12704 12705 12706 12707 12708 12709 12710 12711 12712 12713 12714
2 1 1 1 1 1 1 1 1 1 1 2 1
12715 12716 12717 12718 12719 12720 12721 12722 12723 12724 12725 12726 12727
1 4 1 1 1 1 1 1 1 1 1 1 1
12728 12729 12730 12731 12732 12733 12734 12735 12736 12737 12738 12739 12740
1 1 1 1 1 1 1 1 1 1 1 1 3
12741 12742 12743 12744 12745 12746 12747 12748 12749 12750 12751 12752 12753
1 1 1 1 1 1 1 1 1 1 1 1 1
12754 12755 12756 12757 12758 12759 12760 12761 12762 12763 12764 12765 12766
1 1 1 1 4 1 1 1 1 1 1 1 3
12767 12768 12769 12770 12771 12772 12773 12774 12775 12776 12777 12778 12779
1 1 1 1 1 1 1 5 1 1 1 1 1
12780 12781 12782 12783 12784 12785 12786 12787 12788 12789 12790 12791 12792
1 1 1 1 1 1 1 1 1 1 1 2 1
12793 12794 12795 12796 12797 12798 12799 12800 12801 12802 12803 12804 12805
1 1 1 1 1 1 1 1 1 1 1 2 1
12806 12807 12808 12809 12810 12811 12812 12813 12814 12815 12816 12817 12818
2 1 2 1 3 3 1 1 1 1 1 1 1
12819 12820 12821 12822 12823 12824 12825 12826 12827 12828 12829 12830 12831
1 1 1 1 1 1 1 1 2 1 1 1 1
12832 12833 12834 12835 12836 12837 12838 12839 12840 12841 12842 12843 12844
1 1 1 1 1 1 1 1 1 1 2 6 1
12845 12846 12847 12848 12849 12850 12851 12852 12853 12854 12855 12856 12857
1 1 1 1 1 1 1 1 1 1 1 1 1
12858 12859 12860 12861 12862 12863 12864 12865 12866 12867 12868 12869 12870
1 1 1 1 2 1 2 2 1 3 1 1 1
12871 12872 12873 12874 12875 12876 12877 12878 12879 12880 12881 12882 12883
1 1 1 4 1 1 1 1 5 4 1 1 1
12884 12885 12886 12887 12888 12889 12890 12891 12892 12893 12894 12895 12896
1 6 1 1 1 1 1 1 1 1 1 6 1
12897 12898 12899 12900 12901 12902 12903 12904 12905 12906 12907 12908 12909
1 1 1 1 1 1 1 1 1 1 1 1 2
12910 12911 12912 12913 12914 12915 12916 12917 12918 12919 12920 12921 12922
3 1 1 1 1 1 1 1 1 1 1 3 1
12923 12924 12925 12926 12927 12928 12929 12930 12931 12932 12933 12934 12935
2 1 1 1 1 1 1 1 3 1 1 1 1
12936 12937 12938 12939 12940 12941 12942 12943 12944 12945 12946 12947 12948
1 1 1 1 1 1 1 1 1 1 1 1 1
12949 12950 12951 12952 12953 12954 12955 12956 12957 12958 12959 12960 12961
1 1 1 1 1 2 1 1 1 1 1 1 1
12962 12963 12964 12965 12966 12967 12968 12969 12970 12971 12972 12973 12974
1 1 1 1 1 1 1 1 1 1 2 1 1
12975 12976 12977 12978 12979 12980 12981 12982 12983 12984 12985 12986
1 1 1 2 1 1 1 1 1 1 1 1 sum(multiplicity(bmracc_ppp)>1)[1] 2293There are over 2293 duplicates. In this case, it means we have 2293 overlapping data points.
Using the rjitter() function, we deal with these data points by spacing them out slightly around the point. The function adds noise to the data and spreads these points out a little to facilitate visualization.
bmr_ppp_jit <- rjitter(bmracc_ppp,
retry=TRUE,
nsim=1,
drop=TRUE)Creating an owin object
When analyzing Spatial Point patterns, it is a good practice to confine the boundaries of the analysis with a Geographical area, such as Singapore’s boundary. Using the spatstat package, we can create an owin object that is designed to represent such polygonal regions.
We use the as.owin() function as shown in the code chunk below.
bmr_owin <- as.owin(bmr_boundary)plot(bmr_owin)
Computing KDE using automatic bandwidth selection
We use the density() function of the spatstat package to compute the kernel density. These are the key configurations used in the computation:
bw.diggle() automatic bandwidth selection method.
The smoothing kernel used in this instance is gaussian. Other smoothing methods include “epanechnikov”, “quartic”. or “disc”.
kde_bmracc_bw <- density(bmr_ppp_jit,
sigma=bw.diggle,
edge=TRUE,
kernel="gaussian") The other two methods aside from bw.diggle() are bw.scott() and bw.ppl(). While bw.diggle() focuses on minimizing error in spatial density estimation for point process data and is tailor-made for spatial applications, bw.scott() (Scott’s rule) provides a rule-of-thumb bandwidth and is used in several KDE applications across different types of data besides just spatial data. bw.ppl() uses a more complex and data-driven approach (plug-in) for selecting the bandwidth, aiming to minimize the error in KDE. Like bw.diggle(), It is also tailor-made for spatial point processes, however, it takes a slightly different approach to bw.diggle()
We will now plot the above using the plot() function.
plot(kde_bmracc_bw)
Immediately we notice that the density values are very small. This is because the unit of measurement is in meter, meaning that the density values are computed with a unit of ‘number of points per square meter.’
We now use the rescale.ppp() function of the spatstat package to convert the unit of measurement from meter to kilometer. This is done by implementing the code chunk below.
bmracc_ppp.km <- rescale.ppp(bmr_ppp_jit, 1000, "km")After this, we re-deploy the density() function using the re-scaled data and plot the KDE map.
kde_bmracc.bw <- density(bmracc_ppp.km,
sigma=bw.diggle,
edge=TRUE,
kernel="gaussian")
plot(kde_bmracc.bw)
Converting the KDE output into a grid object
Gridded raster
We convert the gridded kernel density objects into RasterLayer object by using raster() of the raster package.
kde_bmracc_bw_raster <- raster(kde_bmracc.bw)The CRS above is NA, so we now set the CRS to EPSG 32647 of Thailand.
After that, we can implement the plot() function to plot the same.
projection(kde_bmracc_bw_raster) <- CRS("+init=EPSG:32647")
plot(kde_bmracc_bw_raster)
From the above, we infer that the south-east region of Bangkok Metropolitan Region, Samut Prakan Province, specifically has a few hotspots, we want to further analyze the cause of this, so we filter down our data.
smt_prk_bndry=bmr_boundary%>%filter(ADM1_EN=='Samut Prakan')
qtm(smt_prk_bndry)
further eda
# Switch to interactive mode (optional)
tmap_mode("view")
tm_shape(kde_bmracc_bw_raster) +
tm_raster(palette = "-RdYlGn", style = "cont", title = "Accident Density", midpoint = NA,
colorNA = NULL) + # colorNA = NULL will make NAs transparent
tm_shape(bmr_roads) +
tm_lines(col = "black", lwd = 1) + # Overlay the roads
tm_layout(legend.outside = TRUE)bmr_speedingacc=bmr_accsf%>%filter(presumed_cause=='speeding')# Set tmap to interactive view mode (optional for exploration)
tmap_mode("view")
# Create the map with boundaries and accident points
tm_shape(bmr_boundary) +
tm_borders(col = "black") + # Show the boundaries
tm_shape(bmr_speedingacc) +
tm_dots(col = "red", size = 0.01, popup.vars = c("Accident Cause" = "presumed_cause", "Hour of day"="hourofday"), # Add cause to popup
title = "Accidents") + # Show accident points as red dots
tm_layout(frame = FALSE, legend.outside = TRUE)# Switch back to static mode after rendering (if needed)
tmap_mode("plot")# Assuming roads have a column for road type, such as 'road_type'
accidents_on_roads <- st_join(bmr_accsf, bmr_roads, join = st_intersects)
# View the result
head(accidents_on_roads)Simple feature collection with 6 features and 51 fields
Geometry type: POINT
Dimension: XY
Bounding box: xmin: 627012.3 ymin: 1502876 xmax: 693488.9 ymax: 1533381
Projected CRS: WGS 84 / UTM zone 47N
# A tibble: 6 × 52
acc_code incident_datetime report_datetime province_th province_en
<dbl> <dttm> <dttm> <chr> <chr>
1 571882 2019-01-01 02:25:00 2019-01-02 17:32:00 นครปฐม Nakhon Pathom
2 600001 2019-01-01 03:00:00 2019-01-05 10:33:00 นนทบุรี Nonthaburi
3 605043 2019-01-01 03:00:00 2019-03-29 08:22:00 สมุทรปราการ Samut Prakan
4 629691 2019-01-01 03:05:00 2019-01-01 03:05:00 กรุงเทพมหานคร Bangkok
5 571887 2019-01-01 04:30:00 2019-01-02 17:32:00 นครปฐม Nakhon Pathom
6 599234 2019-01-01 04:45:00 2019-01-02 08:28:00 สมุทรปราการ Samut Prakan
# ℹ 47 more variables: agency <chr>, route <chr>, vehicle_type <chr>,
# presumed_cause <chr>, accident_type <chr>,
# number_of_vehicles_involved <dbl>, number_of_fatalities <dbl>,
# number_of_injuries <dbl>, weather_condition <chr>, road_description <chr>,
# slope_description <chr>, geometry <POINT [m]>, Month <dbl>,
# Month_fac <ord>, dayofmonth <int>, hourofday <int>, traffic_period <chr>,
# Shape_Leng <dbl>, Shape_Area <dbl>, ADM1_EN <chr>, ADM1_TH <chr>, …# Count accidents by road type
accidents_by_road_type <- accidents_on_roads %>%
group_by(vehicle_type) %>%
summarise(accident_count = n())%>%
arrange(desc(accident_count))%>%
slice_head(n=5)
# Bar plot showing the number of accidents by road type
ggplot(accidents_by_road_type, aes(x = vehicle_type, y = accident_count, fill = vehicle_type)) +
geom_bar(stat = "identity") +
labs(title = "Number of Accidents by Road Type", x = "Vehicle Type", y = "Accident Count") +
theme_minimal() +
theme(legend.position = "none")
Second-Order Spatial Point Pattern Analysis.
We now focus on Second Order Spatial Point Patterns Analysis.
We first filter the data down to the regions of interest and create separate boundaries for each of them. Following this, we produce an owin object using the as.owin() function.
bkk <- bmr_boundary %>%
filter(ADM1_EN == "Bangkok")
smt_pkn <- bmr_boundary %>%
filter(ADM1_EN == "Samut Prakan")
ntbr <- bmr_boundary %>%
filter(ADM1_EN == "Nonthaburi")
nkn_ptn<- bmr_boundary %>%
filter(ADM1_EN == "Nakhon Pathom")
smt_skn<- bmr_boundary %>%
filter(ADM1_EN == "Samut Sakhon")
bkk_owin = as.owin(bkk)
smt_pkn_owin = as.owin(smt_pkn)
ntbr_owin = as.owin(ntbr)
nkn_ptn_owin = as.owin(nkn_ptn)
smt_skn_owin = as.owin(smt_skn)
acc_bkk_ppp = bmr_ppp_jit[bkk_owin]
acc_smt_pkn_ppp = bmr_ppp_jit[smt_pkn_owin]
acc_ntbr_ppp = bmr_ppp_jit[ntbr_owin]
acc_nkn_ptn_ppp = bmr_ppp_jit[nkn_ptn_owin]
acc_smt_skn_ppp= bmr_ppp_jit[smt_skn_owin]Calculating G-function estimates and testing for Complete Spatial Randomness
We use the Gest() function of the spatstat package to compute a G-function estimation for Bangkok. Following that, we plot the result.
G_bkk = Gest(acc_bkk_ppp, correction = "border")
plot(G_bkk, xlim=c(0,500))
We now carry out the monte carlo simulation test for complete spatial randomness.
G_bkk.csr <- envelope(acc_bkk_ppp, Gest, nsim = 99)Generating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99.
Done.plot(G_bkk.csr)
It is clear that that there is clustering exhibited in this scenario as the black line, observed values, is far above the envelope. We have sufficient evidence to reject the null hypothesis.
We use the Gest() function of the spatstat package to compute a G-function estimation for Samut Prakhan. Following that, we plot the result.
G_smt_pkn = Gest(acc_smt_pkn_ppp, correction = "border")
plot(G_smt_pkn, xlim=c(0,500))
We now implement a monte carlo simulation test for Complete Spatial Random in Samut Prakan.
G_smt_pkn.csr <- envelope(acc_smt_pkn_ppp, Gest, nsim = 199)Generating 199 simulations of CSR ...
1, 2, 3, 4.6.8.10.12.14.16.18.20.22.24.26.28.30.32.34
.36.38.40.42.44.46.48.50.52.54.56.58.60.62.64.66.68.70.72.74
.76.78.80.82.84.86.88.90.92.94.96.98.100.102.104.106.108.110.112.114
.116.118.120.122.124.126.128.130.132.134.136.138.140.142.144.146.148.150.152.154
.156.158.160.162.164.166.168.170.172.174.176.178.180.182.184.186.188.190.192.194
.196.198
199.
Done.plot(G_smt_pkn.csr)
We once again have sufficient evidence to reject the null hypothesis and conclude that accidents in Samut Prakan do indeed exhibit clustering, more so than is expected in a region that is randomly distributed.
We use the Gest() function of the spatstat package to compute a G-function estimation for Nonthaburi. Following that, we plot the result.
G_ntbr = Gest(acc_ntbr_ppp, correction = "border")
plot(G_ntbr, xlim=c(0,500))
We now implement a Monte-Carlo Simulation test to check for complete spatial randomness of roads accidents in Nonthaburi.
G_ntbr.csr <- envelope(acc_ntbr_ppp, Gest, nsim = 199)Generating 199 simulations of CSR ...
1, 2, 3, 4.6.8.10.12.14.16.18.20.22.24.26.28.30.32.34
.36.38.40.42.44.46.48.50.52.54.56.58.60.62.64.66.68.70.72.74
.76.78.80.82.84.86.88.90.92.94.96.98.100.102.104.106.108.110.112.114
.116.118.120.122.124.126.128.130.132.134.136.138.140.142.144.146.148.150.152.154
.156.158.160.162.164.166.168.170.172.174.176.178.180.182.184.186.188.190.192.194
.196.198
199.
Done.plot(G_ntbr.csr)
We use the Gest() function of the spatstat package to compute a G-function estimation for Nakhon Pathom. Following that, we plot the result.
G_nkn_ptn = Gest(acc_nkn_ptn_ppp, correction = "border")
plot(G_nkn_ptn, xlim=c(0,500))
We now conduct the Monte-Carlo Simulation test for complete spatial randomness for accidents in Nakhon Pathom.
G_nkn_ptn.csr <- envelope(acc_nkn_ptn_ppp, Gest, nsim = 199)Generating 199 simulations of CSR ...
1, 2, 3, 4.6.8.10.12.14.16.18.20.22.24.26.28.30.32.34
.36.38.40.42.44.46.48.50.52.54.56.58.60.62.64.66.68.70.72.74
.76.78.80.82.84.86.88.90.92.94.96.98.100.102.104.106.108.110.112.114
.116.118.120.122.124.126.128.130.132.134.136.138.140.142.144.146.148.150.152.154
.156.158.160.162.164.166.168.170.172.174.176.178.180.182.184.186.188.190.192.194
.196.198
199.
Done.plot(G_nkn_ptn.csr)
We use the Gest() function of the spatstat package to compute a G-function estimation for Samut Sakhon. Following that, we plot the result.
G_smt_skn = Gest(acc_smt_skn_ppp, correction = "border")
plot(G_smt_skn, xlim=c(0,500))
We now conduct the Monte-Carlo Simulation test for complete spatial randomness for accidents in Samut Sakhon.
G_smt_skn.csr <- envelope(acc_smt_skn_ppp, Gest, nsim = 199)Generating 199 simulations of CSR ...
1, 2, 3, 4.6.8.10.12.14.16.18.20.22.24.26.28.30.32.34
.36.38.40.42.44.46.48.50.52.54.56.58.60.62.64.66.68.70.72.74
.76.78.80.82.84.86.88.90.92.94.96.98.100.102.104.106.108.110.112.114
.116.118.120.122.124.126.128.130.132.134.136.138.140.142.144.146.148.150.152.154
.156.158.160.162.164.166.168.170.172.174.176.178.180.182.184.186.188.190.192.194
.196.198
199.
Done.plot(G_smt_skn.csr)
bmr_districts=st_intersection(map2_sf, bmr_roads)write_rds(bmr_districts, "data/rds/bmr_districts_roads")bmr_districts=read_rds("data/rds/bmr_districts_roads")# Set tmap to interactive view mode (optional for exploration)
tmap_mode("plot")
# Create the map with boundaries and accident points
tm_shape(bmr_boundary2) +
tm_borders(col = "black") + # Show the boundaries
tm_shape(bmr_accsf) +
tm_dots(col = "red", size = 0.01) + # Show accident points as red dots
tm_layout(frame = FALSE, legend.outside = TRUE)
# Perform a spatial join to assign accidents to districts
accidents_with_districts <- st_join(bmr_accsf, bmr_boundary2, join = st_intersects)
# Group by adm2_en and count the number of accidents per district
accident_counts_by_adm2 <- accidents_with_districts %>%
group_by(ADM2_EN) %>%
summarise(accident_count = n())
# View the resulting dataframe
print(accident_counts_by_adm2)Simple feature collection with 73 features and 2 fields
Geometry type: GEOMETRY
Dimension: XY
Bounding box: xmin: 591277.5 ymin: 1486846 xmax: 710166.1 ymax: 1576520
Projected CRS: WGS 84 / UTM zone 47N
# A tibble: 73 × 3
ADM2_EN accident_count geometry
<chr> <int> <MULTIPOINT [m]>
1 Ban Phaeo 110 ((613054.2 1498274), (615459.1 1494000), (615…
2 Bang Bo 417 ((696848.1 1494239), (696911.6 1494215), (697…
3 Bang Bon 100 ((647897.3 1507733), (647984.3 1507801), (648…
4 Bang Bua Thong 238 ((644467.4 1538181), (645940.6 1538078), (646…
5 Bang Kapi 10 ((673251.8 1520460), (673253.2 1520520), (673…
6 Bang Khae 320 ((650262.2 1517270), (651917.2 1516322), (651…
7 Bang Khen 126 ((671860.4 1534927), (672097.5 1534812), (672…
8 Bang Kho Laem 13 ((661444.6 1515089), (661488.6 1515062), (661…
9 Bang Khun Thian 273 ((649010.8 1505014), (649042.5 1505674), (649…
10 Bang Kruai 50 ((643844.5 1527423), (644559.8 1526978), (645…
# ℹ 63 more rowsSpatial Weights
# Perform a left join with the boundary data
boundary_with_accident_count <- bmr_boundary2 %>%
st_join(accident_counts_by_adm2, by = "ADM2_EN")
boundary_with_accident_count <- boundary_with_accident_count %>%
mutate(accident_count = ifelse(is.na(accident_count), 0, accident_count))
write_rds(boundary_with_accident_count, "data/rds/boundary_with_acc_count")boundary_with_accident_count=read_rds("data/rds/boundary_with_acc_count")basemap <- tm_shape(boundary_with_accident_count) +
tm_polygons() +
tm_text("ADM2_EN.x", size=0.5)
gdppc <- qtm(boundary_with_accident_count, "accident_count")
tmap_arrange(basemap, gdppc, asp=1, ncol=2)
wm_q <- poly2nb(boundary_with_accident_count, queen=TRUE)
summary(wm_q)Neighbour list object:
Number of regions: 79
Number of nonzero links: 438
Percentage nonzero weights: 7.018106
Average number of links: 5.544304
Link number distribution:
2 3 4 5 6 7 8 9
1 5 10 24 20 12 6 1
1 least connected region:
66 with 2 links
1 most connected region:
30 with 9 linkswm_r <- poly2nb(boundary_with_accident_count, queen=FALSE)
summary(wm_r)Neighbour list object:
Number of regions: 79
Number of nonzero links: 422
Percentage nonzero weights: 6.761737
Average number of links: 5.341772
Link number distribution:
2 3 4 5 6 7 8
1 5 16 20 22 11 4
1 least connected region:
66 with 2 links
4 most connected regions:
1 5 17 54 with 8 linkslongitude <- map_dbl(boundary_with_accident_count$geometry, ~st_centroid(.x)[[1]])latitude <- map_dbl(boundary_with_accident_count$geometry, ~st_centroid(.x)[[2]])coords <- cbind(longitude, latitude)head(coords) longitude latitude
[1,] 661951.4 1521172
[2,] 664121.1 1523923
[3,] 700735.2 1532207
[4,] 664808.5 1518057
[5,] 676193.5 1533602
[6,] 677305.1 1522820par(mfrow=c(1,2))
plot(boundary_with_accident_count$geometry, border="lightgrey", main="Queen Contiguity")
plot(wm_q, coords, pch = 19, cex = 0.6, add = TRUE, col= "purple")
plot(boundary_with_accident_count$geometry, border="lightgrey", main="Rook Contiguity")
plot(wm_r, coords, pch = 19, cex = 0.6, add = TRUE, col = "purple")
Adaptive Distance Weight Matrix
knn6 <- knn2nb(knearneigh(coords, k=6))
knn6Neighbour list object:
Number of regions: 79
Number of nonzero links: 474
Percentage nonzero weights: 7.594937
Average number of links: 6
Non-symmetric neighbours listplot(boundary_with_accident_count$geometry, border="lightgrey")
plot(knn6, coords, pch = 19, cex = 0.6, add = TRUE, col = "red")
Row standardized weights matrix
Weights Based on Inversed Distance Weighting (IDW)
dist <- nbdists(wm_q, coords, longlat = FALSE)
ids <- lapply(dist, function(x) 1/(x))
ids[[1]]
[1] 0.0002854535 0.0006582134 0.0004566217 0.0002167536 0.0003080119
[6] 0.0002757322 0.0003371576 0.0002806965
[[2]]
[1] 0.0002854535 0.0002055287 0.0003012437 0.0003439200 0.0003703229
[6] 0.0002193608 0.0001383976 0.0002982074
[[3]]
[1] 8.230456e-05 7.194802e-05 7.691011e-05 6.386761e-05
[[4]]
[1] 0.0005397094 0.0003348823 0.0004742967 0.0004145926 0.0006228750
[[5]]
[1] 1.466707e-04 1.197978e-04 1.317261e-04 1.901079e-04 1.449448e-04
[6] 2.005737e-04 1.397857e-04 8.481219e-05
[[6]]
[1] 0.0001622765 0.0002212846 0.0002009303 0.0001431074 0.0001781389
[6] 0.0002901167
[[7]]
[1] 0.0002055287 0.0005397094 0.0003428497 0.0003432013 0.0003504051
[6] 0.0002113326 0.0005031250 0.0001878859
[[8]]
[1] 0.0006582134 0.0003012437 0.0003348823 0.0003428497 0.0007193454
[[9]]
[1] 0.0001452881 0.0001919620 0.0002478517 0.0001846853 0.0003608347
[6] 0.0001275280
[[10]]
[1] 8.230456e-05 1.204212e-04 1.185473e-04 1.177461e-04 1.338999e-04
[[11]]
[1] 7.194802e-05 1.204212e-04 7.412539e-05 8.957108e-05 5.200876e-05
[6] 6.530843e-05 8.333031e-05
[[12]]
[1] 0.0002120327 0.0003601503 0.0003033661 0.0002204848 0.0002299417
[[13]]
[1] 0.0004566217 0.0004742967 0.0003432013 0.0007193454 0.0005352046
[[14]]
[1] 0.0003439200 0.0004723701 0.0002181006 0.0001852359 0.0003556913
[[15]]
[1] 0.0002167536 0.0004178016 0.0005004725 0.0002059056 0.0002024628
[6] 0.0002988328 0.0002534194
[[16]]
[1] 0.0003080119 0.0004178016 0.0001666057 0.0003222415 0.0002506523
[[17]]
[1] 0.0001622765 0.0004355978 0.0001468778 0.0001493430 0.0002022661
[6] 0.0001455745 0.0002525101 0.0003209334
[[18]]
[1] 0.0002757322 0.0004145926 0.0005352046 0.0005004725 0.0003077165
[6] 0.0003335152
[[19]]
[1] 0.0001666057 0.0002462888 0.0001860044 0.0001509068 0.0001330087
[6] 0.0001378191 0.0002092546
[[20]]
[1] 0.0003371576 0.0003222415 0.0002462888 0.0001834292 0.0002732088
[[21]]
[1] 8.969997e-05 1.203769e-04 1.245294e-04 9.569804e-05 4.958860e-05
[[22]]
[1] 0.0002059056 0.0002506523 0.0001860044 0.0001834292 0.0002152303
[6] 0.0001971194 0.0001095917
[[23]]
[1] 2.053579e-04 1.119769e-04 1.692192e-04 6.579843e-05 1.036058e-04
[[24]]
[1] 0.0002120327 0.0002024628 0.0003529966 0.0002347366 0.0001982741
[6] 0.0001621175
[[25]]
[1] 0.0002806965 0.0003703229 0.0001509068 0.0002732088 0.0001935337
[6] 0.0001208318
[[26]]
[1] 0.0004723701 0.0004355978 0.0001754699 0.0003013611
[[27]]
[1] 0.0001466707 0.0002212846 0.0002022599 0.0003179325 0.0001494025
[[28]]
[1] 0.0006228750 0.0003504051 0.0003601503 0.0003077165 0.0003330968
[6] 0.0002241661
[[29]]
[1] 2.193608e-04 2.181006e-04 1.935337e-04 2.480081e-04 1.269610e-04
[6] 1.731795e-04 8.705329e-05
[[30]]
[1] 0.0001383976 0.0001197978 0.0001852359 0.0001468778 0.0001754699
[6] 0.0002480081 0.0002134931 0.0001692547 0.0001150417
[[31]]
[1] 0.0003033661 0.0002988328 0.0003335152 0.0003529966 0.0003330968
[[32]]
[1] 1.452881e-04 7.412539e-05 1.648225e-04 1.385556e-04 1.384360e-04
[6] 9.877756e-05
[[33]]
[1] 0.0002113326 0.0001919620 0.0002204848 0.0002241661 0.0004631083
[6] 0.0001381990
[[34]]
[1] 0.0002009303 0.0002478517 0.0001493430 0.0001648225 0.0002087434
[6] 0.0001252900
[[35]]
[1] 2.534194e-04 8.969997e-05 2.152303e-04 2.347366e-04 1.338459e-04
[6] 1.190158e-04
[[36]]
[1] 1.317261e-04 1.741388e-04 1.477787e-04 9.360890e-05 1.023839e-04
[6] 4.522313e-05
[[37]]
[1] 0.0002982074 0.0005031250 0.0003556913 0.0002022661 0.0003013611
[6] 0.0001760696
[[38]]
[1] 0.0001901079 0.0001431074 0.0001455745 0.0002022599 0.0002134931
[6] 0.0001969405
[[39]]
[1] 0.0001878859 0.0001846853 0.0002525101 0.0004631083 0.0002087434
[6] 0.0001760696
[[40]]
[1] 0.0001330087 0.0001971194 0.0002053579 0.0001396487 0.0001509543
[[41]]
[1] 0.0001449448 0.0001269610 0.0001692547 0.0001741388 0.0001155726
[6] 0.0001062678
[[42]]
[1] 2.005737e-04 1.477787e-04 1.006357e-04 5.914874e-05
[[43]]
[1] 0.0001397857 0.0001185473 0.0003179325 0.0001503042 0.0001120205
[[44]]
[1] 1.781389e-04 1.177461e-04 8.957108e-05 1.494025e-04 1.385556e-04
[6] 1.252900e-04 1.503042e-04
[[45]]
[1] 0.0002901167 0.0003209334 0.0001969405
[[46]]
[1] 7.691011e-05 8.481219e-05 1.338999e-04 1.006357e-04 1.120205e-04
[6] 7.834800e-05
[[47]]
[1] 3.608347e-04 1.384360e-04 8.847607e-05 7.825942e-05 1.380165e-04
[[48]]
[1] 1.378191e-04 1.119769e-04 1.396487e-04 1.390484e-04 5.812114e-05
[6] 9.420908e-05
[[49]]
[1] 0.0001203769 0.0001982741 0.0001338459 0.0001460302 0.0001218403
[[50]]
[1] 1.245294e-04 1.095917e-04 1.692192e-04 1.190158e-04 1.509543e-04
[6] 4.881437e-05 7.827893e-05
[[51]]
[1] 8.847607e-05 4.260505e-05 9.775096e-05 7.211651e-05 6.976002e-05
[[52]]
[1] 5.200876e-05 4.260505e-05 6.461671e-05 1.208164e-04
[[53]]
[1] 6.530843e-05 9.877756e-05 7.825942e-05 9.775096e-05 6.461671e-05
[6] 1.098016e-04
[[54]]
[1] 1.275280e-04 2.299417e-04 1.621175e-04 1.381990e-04 1.380165e-04
[6] 1.460302e-04 7.211651e-05 8.749710e-05
[[55]]
[1] 9.569804e-05 1.218403e-04 6.976002e-05 8.749710e-05
[[56]]
[1] 8.333031e-05 1.208164e-04 1.098016e-04
[[57]]
[1] 0.0001731795 0.0001150417 0.0001155726 0.0001121332 0.0000764271
[6] 0.0000755279 0.0001320269
[[58]]
[1] 2.092546e-04 1.208318e-04 8.705329e-05 1.390484e-04 1.121332e-04
[6] 1.356813e-04 6.810787e-05
[[59]]
[1] 7.642710e-05 1.356813e-04 1.153320e-04 5.588948e-05 1.070876e-04
[[60]]
[1] 7.552790e-05 1.153320e-04 8.048614e-05 9.201600e-05 7.698649e-05
[[61]]
[1] 5.588948e-05 8.048614e-05 8.676092e-05 7.105006e-05 5.108714e-05
[[62]]
[1] 9.360890e-05 1.062678e-04 1.320269e-04 9.201600e-05 1.156307e-04
[6] 6.087272e-05
[[63]]
[1] 1.023839e-04 1.156307e-04 5.287764e-05 4.047452e-05 6.619029e-05
[6] 3.620195e-05 9.994772e-05
[[64]]
[1] 5.287764e-05 8.616137e-05 5.274538e-05 6.005288e-05
[[65]]
[1] 4.047452e-05 8.616137e-05 5.965942e-05 1.524056e-04
[[66]]
[1] 5.274538e-05 5.965942e-05
[[67]]
[1] 7.698649e-05 8.676092e-05 6.087272e-05 6.619029e-05 7.655740e-05
[[68]]
[1] 6.386761e-05 4.522313e-05 5.914874e-05 7.834800e-05 3.620195e-05
[6] 1.524056e-04
[[69]]
[1] 9.994772e-05 6.005288e-05 7.655740e-05
[[70]]
[1] 4.117989e-05 5.885227e-05 6.155251e-05 4.225974e-05
[[71]]
[1] 4.117989e-05 5.953357e-05 4.110606e-05
[[72]]
[1] 5.885227e-05 5.899804e-05 4.000195e-05 8.692110e-05 8.457300e-05
[[73]]
[1] 6.155251e-05 5.953357e-05 5.899804e-05 6.741263e-05
[[74]]
[1] 7.105006e-05 4.110606e-05 4.000195e-05 6.741263e-05 3.789103e-05
[[75]]
[1] 6.579843e-05 5.812114e-05 4.225974e-05 8.692110e-05 7.172077e-05
[6] 1.056880e-04 5.465069e-05
[[76]]
[1] 9.420908e-05 6.810787e-05 1.070876e-04 5.108714e-05 8.457300e-05
[6] 3.789103e-05 7.172077e-05
[[77]]
[1] 4.958860e-05 4.881437e-05 7.004105e-05 6.324110e-05
[[78]]
[1] 1.036058e-04 7.827893e-05 1.056880e-04 7.004105e-05 5.420721e-05
[[79]]
[1] 5.465069e-05 6.324110e-05 5.420721e-05rswm_q <- nb2listw(wm_q, style="W", zero.policy = TRUE)
rswm_qCharacteristics of weights list object:
Neighbour list object:
Number of regions: 79
Number of nonzero links: 438
Percentage nonzero weights: 7.018106
Average number of links: 5.544304
Weights style: W
Weights constants summary:
n nn S0 S1 S2
W 79 6241 79 29.8143 321.0615Setting the argument zero.policy to TRUE allows for lists of non-neighbors. This should be used with caution as users may not be aware of missing neighbors in their data however setting zero,policy to FALSE would return an error.
The code chunk below is implemented to check the weights of the first polygons eight neighbors type:
rswm_q$weights[10][[1]]
[1] 0.2 0.2 0.2 0.2 0.2Using the same method, we derive a row standardized distance weight matrix by using the code chunk below.
rswm_ids <- nb2listw(wm_q, glist=ids, style="B", zero.policy=TRUE)
rswm_idsCharacteristics of weights list object:
Neighbour list object:
Number of regions: 79
Number of nonzero links: 438
Percentage nonzero weights: 7.018106
Average number of links: 5.544304
Weights style: B
Weights constants summary:
n nn S0 S1 S2
B 79 6241 0.07702966 4.057545e-05 0.0004441685acc.lag <- lag.listw(rswm_q, boundary_with_accident_count$accident_count)
acc.lag [1] 3.375000 42.750000 440.750000 10.800000 141.000000 115.333333
[7] 12.500000 10.600000 222.500000 400.600000 351.714286 123.000000
[13] 10.800000 39.600000 30.714286 90.400000 58.375000 5.166667
[19] 162.428571 94.400000 239.600000 144.857143 299.600000 136.333333
[25] 84.333333 73.500000 206.400000 41.500000 72.142857 57.222222
[31] 40.800000 350.000000 116.333333 163.333333 76.333333 203.333333
[37] 27.833333 60.166667 53.833333 258.000000 119.166667 240.250000
[43] 157.400000 311.428571 34.333333 241.000000 366.800000 190.666667
[49] 212.000000 234.857143 324.200000 540.500000 444.500000 86.875000
[55] 261.750000 626.333333 112.571429 223.857143 115.200000 124.600000
[61] 120.200000 120.000000 290.428571 149.750000 348.000000 523.000000
[67] 135.800000 165.500000 291.000000 121.500000 125.333333 140.600000
[73] 133.250000 67.600000 190.285714 206.000000 138.250000 262.400000
[79] 390.666667nb1 <- wm_q[[1]]
nb1 <- boundary_with_accident_count$accident_count[nb1]
nb1[1] 0 0 0 2 0 1 1 23lag.list <- list(boundary_with_accident_count$ADM2_EN.x, lag.listw(rswm_q, boundary_with_accident_count$accident_count))
lag.res <- as.data.frame(lag.list)
colnames(lag.res) <- c("ADM2_EN.x", "lag Accident Count")
boundary_with_accident_count <- left_join(boundary_with_accident_count,lag.res)acc_qtm <- qtm(boundary_with_accident_count, "accident_count")
lag_acc <- qtm(boundary_with_accident_count, "lag Accident Count")
tmap_arrange(acc_qtm, lag_acc, asp=1, ncol=2)
Spatial Window Average
The spatial window average uses row-standardized weights and includes the diagonal element. To do this in R, we need to go back to the neighbors structure and add the diagonal element before assigning weights.
To add the diagonal element to the neighbour list, we use the include.self() function from the spdep package.
wm_qs <- include.self(wm_q)
wm_qsNeighbour list object:
Number of regions: 79
Number of nonzero links: 517
Percentage nonzero weights: 8.283929
Average number of links: 6.544304 There is a difference in the key statistics shown above when compared to wm_q. The average number of links, the number of non-zero links as well as percentage of non-zero weights are all higher for wm_qs.
We look at the neighbor list of area [1] using the code chunk below.
wm_qs[[1]][1] 1 2 8 13 15 16 18 20 25This region has 9 neighbors.
We now implement the nb2listw() function in order to obtain weights.
wm_qs <- nb2listw(wm_qs)
wm_qsCharacteristics of weights list object:
Neighbour list object:
Number of regions: 79
Number of nonzero links: 517
Percentage nonzero weights: 8.283929
Average number of links: 6.544304
Weights style: W
Weights constants summary:
n nn S0 S1 S2
W 79 6241 79 24.99173 318.4378We now create the lag variable from our weight structure and accident_count variable.
lag_w_avg_acc <- lag.listw(wm_qs,
boundary_with_accident_count$accident_count)
lag_w_avg_acc [1] 3.000000 38.000000 364.600000 10.333333 139.333333 100.285714
[7] 16.111111 8.833333 191.571429 345.833333 423.500000 125.500000
[13] 9.000000 46.666667 27.125000 75.333333 55.000000 4.571429
[19] 198.000000 78.833333 245.166667 127.000000 249.666667 124.714286
[25] 75.571429 60.000000 174.833333 36.714286 64.125000 66.500000
[31] 36.166667 377.714286 106.000000 163.714286 86.000000 184.285714
[37] 28.714286 60.857143 47.000000 268.333333 106.571429 225.000000
[43] 205.333333 320.750000 33.250000 233.142857 319.000000 269.285714
[49] 181.000000 218.000000 312.333333 515.800000 457.571429 132.222222
[55] 228.000000 581.500000 119.625000 202.125000 115.833333 143.500000
[61] 110.000000 130.285714 270.125000 252.000000 355.400000 356.666667
[67] 127.166667 224.571429 233.750000 146.200000 114.500000 134.833333
[73] 112.800000 73.000000 199.875000 187.750000 277.600000 230.333333
[79] 320.500000We then proceed to convert the lag variable listwobject into a data-frame by using as.data.frame().
lag.list.wm_qs <- list(boundary_with_accident_count$ADM2_EN.x, lag.listw(wm_qs, boundary_with_accident_count$accident_count))
lag_wm_qs.res <- as.data.frame(lag.list.wm_qs)
colnames(lag_wm_qs.res) <- c("ADM2_EN.x", "lag_window_avg acc")Note: The third command line on the code chunk above renames the field names of lag_wm_q1.res object into accident_count and lag_window_avg acc respectively.
We now append the lag_window_avg acc values onto the boundary_with_accident_count sf data.frame by using left_join() of dplyr package.
boundary_with_accident_count <- left_join(boundary_with_accident_count, lag_wm_qs.res)To compare the values of lag accident count and Spatial window average, The kable() function of the Knitr package is used to prepare a table.
boundary_with_accident_count %>%
dplyr::select("ADM2_EN.x",
"lag Accident Count",
"lag_window_avg acc") %>%
kable()| ADM2_EN.x | lag Accident Count | lag_window_avg acc | geometry |
|---|---|---|---|
| Phra Nakhon | 3.375000 | 3.000000 | MULTIPOLYGON (((662263.2 15… |
| Dusit | 42.750000 | 38.000000 | MULTIPOLYGON (((664304.4 15… |
| Nong Chok | 440.750000 | 364.600000 | MULTIPOLYGON (((706774.6 15… |
| Bang Rak | 10.800000 | 10.333333 | MULTIPOLYGON (((664040.2 15… |
| Bang Khen | 141.000000 | 139.333333 | MULTIPOLYGON (((673966.4 15… |
| Bang Kapi | 115.333333 | 100.285714 | MULTIPOLYGON (((676080.6 15… |
| Pathum Wan | 12.500000 | 16.111111 | MULTIPOLYGON (((664236.5 15… |
| Pom Prap Sattru Phai | 10.600000 | 8.833333 | MULTIPOLYGON (((663880.5 15… |
| Phra Khanong | 222.500000 | 191.571429 | MULTIPOLYGON (((674748.8 15… |
| Min Buri | 400.600000 | 345.833333 | MULTIPOLYGON (((694735.5 15… |
| Lat Krabang | 351.714286 | 423.500000 | MULTIPOLYGON (((694497.3 15… |
| Yan Nawa | 123.000000 | 125.500000 | MULTIPOLYGON (((667818.3 15… |
| Samphanthawong | 10.800000 | 9.000000 | MULTIPOLYGON (((663592.4 15… |
| Phaya Thai | 39.600000 | 46.666667 | MULTIPOLYGON (((667624 1525… |
| Thon Buri | 30.714286 | 27.125000 | MULTIPOLYGON (((661364.9 15… |
| Bangkok Yai | 90.400000 | 75.333333 | MULTIPOLYGON (((660991.6 15… |
| Huai Khwang | 58.375000 | 55.000000 | MULTIPOLYGON (((671695.6 15… |
| Khlong San | 5.166667 | 4.571429 | MULTIPOLYGON (((662113.2 15… |
| Taling Chan | 162.428571 | 198.000000 | MULTIPOLYGON (((654587 1526… |
| Bangkok Noi | 94.400000 | 78.833333 | MULTIPOLYGON (((659654.1 15… |
| Bang Khun Thian | 239.600000 | 245.166667 | MULTIPOLYGON (((656541 1512… |
| Phasi Charoen | 144.857143 | 127.000000 | MULTIPOLYGON (((654370 1519… |
| Nong Khaem | 299.600000 | 249.666667 | MULTIPOLYGON (((647001.3 15… |
| Rat Burana | 136.333333 | 124.714286 | MULTIPOLYGON (((661132.1 15… |
| Bang Phlat | 84.333333 | 75.571429 | MULTIPOLYGON (((663052.9 15… |
| Din Daeng | 73.500000 | 60.000000 | MULTIPOLYGON (((670238.2 15… |
| Bueng Kum | 206.400000 | 174.833333 | MULTIPOLYGON (((678184.5 15… |
| Sathon | 41.500000 | 36.714286 | MULTIPOLYGON (((667917.7 15… |
| Bang Sue | 72.142857 | 64.125000 | MULTIPOLYGON (((666275.4 15… |
| Chatuchak | 57.222222 | 66.500000 | MULTIPOLYGON (((671471.4 15… |
| Bang Kho Laem | 40.800000 | 36.166667 | MULTIPOLYGON (((664335.2 15… |
| Prawet | 350.000000 | 377.714286 | MULTIPOLYGON (((680695.6 15… |
| Khlong Toei | 116.333333 | 106.000000 | MULTIPOLYGON (((673075.4 15… |
| Suan Luang | 163.333333 | 163.714286 | MULTIPOLYGON (((677751.8 15… |
| Chom Thong | 76.333333 | 86.000000 | MULTIPOLYGON (((658696.5 15… |
| Don Mueang | 203.333333 | 184.285714 | MULTIPOLYGON (((674339.8 15… |
| Ratchathewi | 27.833333 | 28.714286 | MULTIPOLYGON (((667199.4 15… |
| Lat Phrao | 60.166667 | 60.857143 | MULTIPOLYGON (((673842.9 15… |
| Vadhana | 53.833333 | 47.000000 | MULTIPOLYGON (((668704.8 15… |
| Bang Khae | 258.000000 | 268.333333 | MULTIPOLYGON (((647693 1520… |
| Lak Si | 119.166667 | 106.571429 | MULTIPOLYGON (((671045.4 15… |
| Sai Mai | 240.250000 | 225.000000 | MULTIPOLYGON (((679686.2 15… |
| Khan Na Yao | 157.400000 | 205.333333 | MULTIPOLYGON (((680806.2 15… |
| Saphan Sung | 311.428571 | 320.750000 | MULTIPOLYGON (((685556.9 15… |
| Wang Thonglang | 34.333333 | 33.250000 | MULTIPOLYGON (((671782.5 15… |
| Khlong Sam Wa | 241.000000 | 233.142857 | MULTIPOLYGON (((693668.9 15… |
| Bang Na | 366.800000 | 319.000000 | MULTIPOLYGON (((675501.8 15… |
| Thawi Watthana | 190.666667 | 269.285714 | MULTIPOLYGON (((650687.8 15… |
| Thung Khru | 212.000000 | 181.000000 | MULTIPOLYGON (((660699.8 15… |
| Bang Bon | 234.857143 | 218.000000 | MULTIPOLYGON (((656343.6 15… |
| Mueang Samut Prakan | 324.200000 | 312.333333 | MULTIPOLYGON (((673201.8 15… |
| Bang Bo | 540.500000 | 515.800000 | MULTIPOLYGON (((712294.1 15… |
| Bang Phli | 444.500000 | 457.571429 | MULTIPOLYGON (((687139.8 15… |
| Phra Pradaeng | 86.875000 | 132.222222 | MULTIPOLYGON (((670827.4 15… |
| Phra Samut Chedi | 261.750000 | 228.000000 | MULTIPOLYGON (((669191.5 15… |
| Bang Sao Thong | 626.333333 | 581.500000 | MULTIPOLYGON (((695481.3 15… |
| Mueang Nonthaburi | 112.571429 | 119.625000 | MULTIPOLYGON (((662112.2 15… |
| Bang Kruai | 223.857143 | 202.125000 | MULTIPOLYGON (((654095.9 15… |
| Bang Yai | 115.200000 | 115.833333 | MULTIPOLYGON (((649983.7 15… |
| Bang Bua Thong | 124.600000 | 143.500000 | MULTIPOLYGON (((649993.4 15… |
| Sai Noi | 120.200000 | 110.000000 | MULTIPOLYGON (((644817.9 15… |
| Pak Kret | 120.000000 | 130.285714 | MULTIPOLYGON (((652378.8 15… |
| Mueang Pathum Thani | 290.428571 | 270.125000 | MULTIPOLYGON (((672162.5 15… |
| Khlong Luang | 149.750000 | 252.000000 | MULTIPOLYGON (((689473.3 15… |
| Thanyaburi | 348.000000 | 355.400000 | MULTIPOLYGON (((706757.7 15… |
| Nong Suea | 523.000000 | 356.666667 | MULTIPOLYGON (((704086 1575… |
| Lat Lum Kaeo | 135.800000 | 127.166667 | MULTIPOLYGON (((650538.8 15… |
| Lam Luk Ka | 165.500000 | 224.571429 | MULTIPOLYGON (((706764.9 15… |
| Sam Khok | 291.000000 | 233.750000 | MULTIPOLYGON (((665864.3 15… |
| Mueang Nakhon Pathom | 121.500000 | 146.200000 | MULTIPOLYGON (((602747 1541… |
| Kamphaeng Saen | 125.333333 | 114.500000 | MULTIPOLYGON (((612428.1 15… |
| Nakhon Chai Si | 140.600000 | 134.833333 | MULTIPOLYGON (((627397.1 15… |
| Don Tum | 133.250000 | 112.800000 | MULTIPOLYGON (((620292.6 15… |
| Bang Len | 67.600000 | 73.000000 | MULTIPOLYGON (((631987.6 15… |
| Sam Phran | 190.285714 | 199.875000 | MULTIPOLYGON (((636176.3 15… |
| Phutthamonthon | 206.000000 | 187.750000 | MULTIPOLYGON (((637713.4 15… |
| Mueang Samut Sakhon | 138.250000 | 277.600000 | MULTIPOLYGON (((646615.4 15… |
| Krathum Baen | 262.400000 | 230.333333 | MULTIPOLYGON (((641549.1 15… |
| Ban Phaeo | 390.666667 | 320.500000 | MULTIPOLYGON (((625054.6 15… |
We now plot the lag accident count and w_ave_acc maps next to each other for comparison using the qtm() function of the tmap package.
w_avg_acc <- qtm(boundary_with_accident_count, "lag_window_avg acc")
tmap_arrange(lag_acc, w_avg_acc, asp=1, ncol=2)
equal <- tm_shape(boundary_with_accident_count) +
tm_fill("accident_count",
n = 5,
style = "equal") +
tm_borders(alpha = 0.5) +
tm_layout(main.title = "Equal interval classification")
quantile <- tm_shape(boundary_with_accident_count) +
tm_fill("accident_count",
n = 5,
style = "quantile") +
tm_borders(alpha = 0.5) +
tm_layout(main.title = "Equal quantile classification")
tmap_arrange(equal,
quantile,
asp=1,
ncol=2)
Spatial Autocorrelation: Morans I test.
The hypotheses for the test are as follows:
H0: Regions with similar number of accidents are randomly distributed.
H1: Regions with similar number of accidents are not randomly distributed and exhibit spatial clustering.
moran.test(boundary_with_accident_count$accident_count,
listw=rswm_q,
zero.policy = TRUE,
na.action=na.omit)
Moran I test under randomisation
data: boundary_with_accident_count$accident_count
weights: rswm_q
Moran I statistic standard deviate = 3.7964, p-value = 7.34e-05
alternative hypothesis: greater
sample estimates:
Moran I statistic Expectation Variance
0.235764728 -0.012820513 0.004287482 From the output above, we can infer the following:
The p-value (7.34e-05)<0.05, indicating that the observed spatial autocorrelation is statistically significant.
Moran’s I statistic: The observed value of 0.236 indicates positive spatial autocorrelation, meaning that regions with similar number of accidents are more likely to be located near each other.
Since Moran’s I Statistic is significantly greater than what we would expect in a randomly distributed region. There is significant evidence to reject H0 and conclude that there is indeed spatial clustering with regards to Accidents in the Bangkok Metropolitan Region.
Monte Carlo Moran’s I
We now implement the moran.mc() function of the spdep package. In this scenario, we will run 1000 simulations.
bperm= moran.mc(boundary_with_accident_count$accident_count,
listw=rswm_q,
nsim=999,
zero.policy = TRUE,
na.action=na.omit)
bperm
Monte-Carlo simulation of Moran I
data: boundary_with_accident_count$accident_count
weights: rswm_q
number of simulations + 1: 1000
statistic = 0.23576, observed rank = 1000, p-value = 0.001
alternative hypothesis: greaterBased on the above output, p-value (0.001)<0.05, thus we can reject the null hypothesis at a 5% significance level and conclude that there is indeed spatial clustering.
Visualizing Monte Carlo Moran’s I
We can visualize the test statistics obtained from the simulation above by implementing the hist() and abline() functions of R graphics.
We first calculate the mean and variance, and obtain the summary statistics.
mean(bperm$res[1:999])[1] -0.01073838var(bperm$res[1:999])[1] 0.00437831summary(bperm$res[1:999]) Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.20513 -0.05612 -0.01518 -0.01074 0.03086 0.23281 hist(bperm$res,
freq=TRUE,
breaks=20,
xlab="Simulated Moran's I")
abline(v=0,
col="red") 
From the above, we can infer that over half of all simulations indicate a negative value for Moran’s I statistic. Generally, a negative value indicates that dissimilar regions are located next to each other. (i.e: regions with dissimilar number of Accidents are located next to each other)
We can also make use of the ggplot2 R package to produce a plot.
data <- data.frame(simulated_moran = bperm$res)
ggplot(data, aes(x = simulated_moran)) +
geom_histogram(binwidth = (max(data$simulated_moran) - min(data$simulated_moran)) / 20,
fill = "lightblue", color = "black") +
geom_vline(xintercept = 0, color = "red", linetype = "dashed") +
labs(x = "Simulated Moran's I",
y = "Frequency",
title = "Histogram of Simulated Moran's I") +
theme_minimal()
Local Morans I
We implement the localmoran() function of spdep compute the local Moran’s I statistic. This function helps us compute li values, given a set of zi values and a listw object providing neighbor weighting information for the polygon associated with the zi values.
We compute local Moran’s I of accident_count at the district level.
fips <- order(boundary_with_accident_count$accident_count)
localMI <- localmoran(boundary_with_accident_count$accident_count, rswm_q)
head(localMI) Ii E.Ii Var.Ii Z.Ii Pr(z != E(Ii))
1 0.60429563 -0.0079097813 0.070446604 2.3065726 0.021078658
2 0.45651029 -0.0079097813 0.070446604 1.7497699 0.080158019
3 -0.65867321 -0.0031893365 0.060342196 -2.6684032 0.007621273
4 0.54837408 -0.0071586128 0.106462820 1.7025922 0.088644413
5 0.02048823 -0.0004311934 0.003869271 0.3363063 0.736639872
6 0.17288609 -0.0069766753 0.085295392 0.6158550 0.537990224localmoran() function returns a matrix of values whose columns are:
Ii: the local Moran’s I statistics
E.Ii: the expectation of local moran statistic under the randomisation hypothesis
Var.Ii: the variance of local moran statistic under the randomisation hypothesis
Z.Ii:the standard deviate of local moran statistic
Pr(): the p-value of local moran statistic
We now use the printCoefmat() to display the content of the local Moran matrix that we created above.
printCoefmat(data.frame(
localMI[fips,],
row.names=boundary_with_accident_count$ADM2_EN.x[fips]),
check.names=FALSE) Ii E.Ii Var.Ii Z.Ii
Phra Nakhon 6.0430e-01 -7.9098e-03 7.0447e-02 2.3066e+00
Dusit 4.5651e-01 -7.9098e-03 7.0447e-02 1.7498e+00
Pom Prap Sattru Phai 5.7718e-01 -7.9098e-03 1.1755e-01 1.7065e+00
Samphanthawong 5.7643e-01 -7.9098e-03 1.1755e-01 1.7044e+00
Bangkok Yai 2.7767e-01 -7.9098e-03 1.1755e-01 8.3295e-01
Nong Khaem -5.0752e-01 -7.9098e-03 1.1755e-01 -1.4572e+00
Khlong San 5.9394e-01 -7.8138e-03 9.5450e-02 1.9477e+00
Bangkok Noi 2.6106e-01 -7.8138e-03 1.1613e-01 7.8898e-01
Thon Buri 4.9558e-01 -7.7185e-03 7.9701e-02 1.7828e+00
Phasi Charoen 7.2382e-02 -7.7185e-03 7.9701e-02 2.8373e-01
Phra Khanong -2.1018e-01 -7.3429e-03 8.9740e-02 -6.7710e-01
Din Daeng 3.2865e-01 -7.3429e-03 1.3835e-01 9.0331e-01
Vadhana 3.9977e-01 -7.3429e-03 8.9740e-02 1.3590e+00
Bang Rak 5.4837e-01 -7.1586e-03 1.0646e-01 1.7026e+00
Sathon 4.3876e-01 -7.1586e-03 8.7504e-02 1.5074e+00
Bang Sue 3.2934e-01 -7.1586e-03 7.3961e-02 1.2373e+00
Bang Kapi 1.7289e-01 -6.9767e-03 8.5295e-02 6.1586e-01
Bang Kho Laem 4.2715e-01 -6.7082e-03 9.9809e-02 1.3733e+00
Bueng Kum -1.4140e-01 -6.3583e-03 9.4637e-02 -4.3898e-01
Bang Phlat 2.5840e-01 -5.8512e-03 7.1616e-02 9.8744e-01
Nong Suea -1.1495e+00 -5.7687e-03 2.2361e-01 -2.4187e+00
Thung Khru -1.5046e-01 -5.6055e-03 8.3495e-02 -5.0131e-01
Huai Khwang 3.3009e-01 -5.4446e-03 4.8612e-02 1.5219e+00
Wang Thonglang 3.9902e-01 -5.2861e-03 1.3487e-01 1.1009e+00
Lak Si 1.3769e-01 -5.2077e-03 6.3782e-02 5.6584e-01
Don Tum 9.4804e-02 -5.2077e-03 9.8330e-02 3.1894e-01
Ratchathewi 4.0649e-01 -4.9761e-03 6.0959e-02 1.6665e+00
Khlong Toei 1.3206e-01 -4.2420e-03 5.2005e-02 5.9770e-01
Pathum Wan 4.1399e-01 -4.1719e-03 3.7296e-02 2.1653e+00
Bang Kruai -1.5533e-01 -3.8297e-03 3.9701e-02 -7.6037e-01
Rat Burana 7.0045e-02 -3.5022e-03 4.2967e-02 3.5481e-01
Sai Noi 1.0630e-01 -3.2507e-03 4.8535e-02 4.9727e-01
Nong Chok -6.5867e-01 -3.1893e-03 6.0342e-02 -2.6684e+00
Phutthamonthon -9.9194e-02 -3.1893e-03 3.3083e-02 -5.2782e-01
Sam Khok -2.9599e-01 -3.0683e-03 7.8458e-02 -1.0458e+00
Lat Phrao 2.3647e-01 -2.8911e-03 3.5491e-02 1.2706e+00
Don Mueang -8.3944e-02 -2.6075e-03 3.2019e-02 -4.5455e-01
Krathum Baen -2.1123e-01 -2.6075e-03 3.8957e-02 -1.0570e+00
Min Buri -4.9826e-01 -2.4982e-03 3.7327e-02 -2.5660e+00
Bang Na -3.8999e-01 -2.0842e-03 3.1155e-02 -2.1977e+00
Phaya Thai 2.3471e-01 -1.9866e-03 2.9698e-02 1.3735e+00
Kamphaeng Saen 7.3445e-02 -1.9866e-03 5.0854e-02 3.3450e-01
Lat Lum Kaeo 5.2453e-02 -1.8913e-03 2.8277e-02 3.2317e-01
Phra Samut Chedi -1.5870e-01 -1.4915e-03 2.8267e-02 -9.3503e-01
Bang Bon -1.0360e-01 -1.2133e-03 1.2611e-02 -9.1175e-01
Bang Len 1.4226e-01 -1.2133e-03 1.8152e-02 1.0649e+00
Nakhon Chai Si 3.1698e-02 -9.9768e-04 1.4930e-02 2.6759e-01
Ban Phaeo -2.8097e-01 -8.6565e-04 2.2184e-02 -1.8806e+00
Bang Yai 5.0958e-02 -6.0283e-04 9.0244e-03 5.4276e-01
Bang Khen 2.0488e-02 -4.3119e-04 3.8693e-03 3.3631e-01
Mueang Pathum Thani -1.0470e-01 -3.8742e-04 4.0301e-03 -1.6432e+00
Yan Nawa 2.4924e-02 -2.0371e-04 3.0508e-03 4.5494e-01
Chom Thong 4.0971e-02 -1.2158e-04 1.4967e-03 1.0622e+00
Chatuchak 3.5183e-02 -6.0530e-05 4.7609e-04 1.6152e+00
Sai Mai -6.5785e-04 -4.2214e-08 8.0124e-07 -7.3488e-01
Suan Luang -3.8712e-05 -7.6848e-07 9.4613e-06 -1.2336e-02
Mueang Nonthaburi -5.4655e-03 -6.2488e-06 6.5027e-05 -6.7699e-01
Khlong Sam Wa 3.7824e-02 -1.3683e-04 1.6844e-03 9.2494e-01
Pak Kret -2.7988e-02 -2.2332e-04 2.7488e-03 -5.2957e-01
Bang Bua Thong -6.6869e-02 -1.5866e-03 2.3728e-02 -4.2380e-01
Mueang Nakhon Pathom -7.8933e-02 -1.9026e-03 3.6044e-02 -4.0574e-01
Mueang Samut Prakan 3.2339e-01 -2.2990e-03 3.4358e-02 1.7571e+00
Sam Phran 6.0701e-02 -3.0827e-03 3.1981e-02 3.5667e-01
Bang Khun Thian 1.8656e-01 -3.4537e-03 5.1556e-02 8.3683e-01
Bang Khae 3.3266e-01 -7.0892e-03 1.0544e-01 1.0463e+00
Thanyaburi 9.2497e-01 -1.4248e-02 2.6658e-01 1.8191e+00
Saphan Sung 7.4410e-01 -1.4378e-02 1.4747e-01 1.9751e+00
Bang Bo 2.1695e+00 -1.8681e-02 3.4795e-01 3.7095e+00
Khan Na Yao -4.4722e-02 -2.3052e-02 3.3734e-01 -3.7310e-02
Taling Chan -1.2591e-02 -2.3382e-02 2.3763e-01 2.2136e-02
Bang Sao Thong 2.9810e+00 -2.3382e-02 5.8570e-01 3.9257e+00
Phra Pradaeng -5.8509e-01 -3.1998e-02 2.7807e-01 -1.0489e+00
Bang Phli 2.3769e+00 -4.0427e-02 4.7760e-01 3.4978e+00
Prawet 1.6089e+00 -4.2186e-02 4.9747e-01 2.3410e+00
Lam Luk Ka 1.0605e-02 -5.0323e-02 5.8839e-01 7.9431e-02
Khlong Luang -1.6589e-01 -7.2197e-02 1.2714e+00 -8.3095e-02
Thawi Watthana 3.4609e-01 -9.7330e-02 1.0817e+00 4.2635e-01
Mueang Samut Sakhon -4.0011e-01 -1.3165e-01 2.1698e+00 -1.8225e-01
Lat Krabang 3.2578e+00 -1.6980e-01 1.4670e+00 2.8299e+00
Pr.z....E.Ii..
Phra Nakhon 0.0211
Dusit 0.0802
Pom Prap Sattru Phai 0.0879
Samphanthawong 0.0883
Bangkok Yai 0.4049
Nong Khaem 0.1451
Khlong San 0.0514
Bangkok Noi 0.4301
Thon Buri 0.0746
Phasi Charoen 0.7766
Phra Khanong 0.4983
Din Daeng 0.3664
Vadhana 0.1741
Bang Rak 0.0886
Sathon 0.1317
Bang Sue 0.2160
Bang Kapi 0.5380
Bang Kho Laem 0.1697
Bueng Kum 0.6607
Bang Phlat 0.3234
Nong Suea 0.0156
Thung Khru 0.6162
Huai Khwang 0.1280
Wang Thonglang 0.2709
Lak Si 0.5715
Don Tum 0.7498
Ratchathewi 0.0956
Khlong Toei 0.5500
Pathum Wan 0.0304
Bang Kruai 0.4470
Rat Burana 0.7227
Sai Noi 0.6190
Nong Chok 0.0076
Phutthamonthon 0.5976
Sam Khok 0.2957
Lat Phrao 0.2039
Don Mueang 0.6494
Krathum Baen 0.2905
Min Buri 0.0103
Bang Na 0.0280
Phaya Thai 0.1696
Kamphaeng Saen 0.7380
Lat Lum Kaeo 0.7466
Phra Samut Chedi 0.3498
Bang Bon 0.3619
Bang Len 0.2869
Nakhon Chai Si 0.7890
Ban Phaeo 0.0600
Bang Yai 0.5873
Bang Khen 0.7366
Mueang Pathum Thani 0.1003
Yan Nawa 0.6492
Chom Thong 0.2882
Chatuchak 0.1063
Sai Mai 0.4624
Suan Luang 0.9902
Mueang Nonthaburi 0.4984
Khlong Sam Wa 0.3550
Pak Kret 0.5964
Bang Bua Thong 0.6717
Mueang Nakhon Pathom 0.6849
Mueang Samut Prakan 0.0789
Sam Phran 0.7213
Bang Khun Thian 0.4027
Bang Khae 0.2954
Thanyaburi 0.0689
Saphan Sung 0.0483
Bang Bo 0.0002
Khan Na Yao 0.9702
Taling Chan 0.9823
Bang Sao Thong 0.0001
Phra Pradaeng 0.2942
Bang Phli 0.0005
Prawet 0.0192
Lam Luk Ka 0.9367
Khlong Luang 0.9338
Thawi Watthana 0.6698
Mueang Samut Sakhon 0.8554
Lat Krabang 0.0047Mapping the local Moran’s I
Before we map the local Moran’s I map, it is wise to append the local Moran’s data-frame (localMI) onto our SpatialPolygonDataFrame.
bmr.localMI <- cbind(boundary_with_accident_count,localMI) %>%
rename(Pr.Ii = Pr.z....E.Ii..)Mapping Local Moran’s I values
We now make use of the tmap package and its choropleth mapping functions to plot the local Moran’s I values.
tm_shape(bmr.localMI) +
tm_fill(col = "Ii",
style = "pretty",
palette = "RdBu",
title = "local moran statistics") +
tm_borders(alpha = 0.5)
Mapping Local Moran’s I p-values
The Choropleth reveals the presence of both positive, as well as negative I values. This indicates that there are varying levels of spatial autocorrelation, however, we must examine the p-values for these I values to check for statistical significance.
We use the tmap package to draw a choropleth map of Moran’s I p-values.
tm_shape(bmr.localMI) +
tm_fill(col = "Pr.Ii",
breaks=c(-Inf, 0.001, 0.01, 0.05, 0.1, Inf),
palette="-Blues",
title = "local Moran's I p-values") +
tm_borders(alpha = 0.5)
Mapping both local Moran’s I values and p-values
In the interest of easier analysis and interpretation, we plot the two maps next to each other.
localMI.map <- tm_shape(bmr.localMI) +
tm_fill(col = "Ii",
style = "pretty",
title = "local moran statistics") +
tm_borders(alpha = 0.5)
pvalue.map <- tm_shape(bmr.localMI) +
tm_fill(col = "Pr.Ii",
breaks=c(-Inf, 0.001, 0.01, 0.05, 0.1, Inf),
palette="-Blues",
title = "local Moran's I p-values") +
tm_borders(alpha = 0.5)
tmap_arrange(localMI.map, pvalue.map, asp=1, ncol=2)
Hot and Cold Spot Areas
Getis and Ord’s G-Statistics
An alternative spatial statistic used to detect spatial anomalies is the Getis-Ord G-statistic (Getis and Ord, 1972; Ord and Getis, 1995). This method examines spatial relationships within a defined proximity to identify clusters of high or low values. Statistically significant hotspots are areas where high values are spatially clustered, meaning that not only do these areas have high values, but their neighboring areas also exhibit similarly high values.
The analysis involves three key steps:
Deriving the spatial weight matrix: This defines the spatial relationships between areas, specifying which locations are considered neighbors based on proximity.
Computing the Gi statistic: This step calculates the G-statistic for each location, identifying regions where values are significantly higher or lower than expected.
Mapping the Gi statistics: The results are visualized to reveal spatial patterns of high-value clusters (hotspots) and low-value clusters (cold spots).
This approach is useful for identifying localized patterns of spatial clustering and detecting significant anomalies in the data.
Deriving Distance Based Weight Matrix
We start by defining a new set of neighbors. While the spatial autocorrelation considered units which shared borders, for Getis-Ord, we will define the neighbors based on distance.
There are two types of distance-based proximity matrices:
Fixed Distance Weight Matrix
Adaptive Distance Weight Matrix
Before creating our connectivity graph, we need to assign a point to each polygon. This requires more than simply running st_centroid() on the us.bound spatial object. Specifically, we need to extract the coordinates into a separate data frame. We have already done this earlier and stored them under the name coords.
Mapping functions apply a specific operation to each element of a vector and return a vector of the same length. In our case, the input vector will be the geometry column from us.bound, and the function we’ll apply is st_centroid(). We’ll use the map_dbl variation from the purrr package, which is designed to return numeric (double) values.
To extract the longitude values, we’ll map the st_centroid() function over the geometry column and use double bracket notation [[]] with 1 to access the first element of each centroid, which corresponds to the longitude.
For more detailed information, you can refer to the map documentation here.
Determine the Cut-off Distance
To determine the upper limit for the distance band, we follow these steps:
Find the k-nearest neighbors: Use the
knearneigh()function from the spdep package. This function returns a matrix that contains the indices of points corresponding to the k-nearest neighbors for each observation.Convert to a neighbors list: Take the k-nearest neighbors object returned by
knearneigh()and convert it into a neighbors list (classnb) by using theknn2nb()function. This generates a list of integer vectors, where each vector contains the region numbers corresponding to its neighbors.Calculate neighbor distances: Use the
nbdists()function from spdep to calculate the distances between neighbors. The function returns the lengths of neighbor relationship edges in the units of the coordinates (e.g., kilometers if the coordinates are geographic).Flatten the distance list: The distances returned by
nbdists()are stored in a list. Use theunlist()function to remove the list structure and return a single vector of distances.
This process helps identify the upper limit for a distance band by analyzing the distances between neighboring regions.
k1 <- knn2nb(knearneigh(coords))
k1dists <- unlist(nbdists(k1, coords, longlat = FALSE))
summary(k1dists) Min. 1st Qu. Median Mean 3rd Qu. Max.
1390 2805 4986 6259 8476 16797 From the above output, we can infer that the largest first-nearest neighbor distance is just under 17000M.
Using this as the upper threshold gives certainty that all units will have at least one neighbor.
Computing fixed distance weight matrix
We implement the dnearneigh() function of the spdep package to compute the distance weight matrix.
wm_d62 <- dnearneigh(coords, 0, 17000, longlat = FALSE)
wm_d62Neighbour list object:
Number of regions: 79
Number of nonzero links: 1766
Percentage nonzero weights: 28.29675
Average number of links: 22.35443 After this, we implement the nb2listw() function to convert the nb object into spatial weights object.
On average, each region has approximately 22.3neighbors.
wm62_lw <- nb2listw(wm_d62, style = 'B')
summary(wm62_lw)Characteristics of weights list object:
Neighbour list object:
Number of regions: 79
Number of nonzero links: 1766
Percentage nonzero weights: 28.29675
Average number of links: 22.35443
Link number distribution:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 19 20 21 22 23 25 26 27 29 31 32
3 5 3 6 1 2 1 2 2 1 1 1 1 1 3 1 1 1 2 1 2 1 1 2 2 1
33 34 35 36 37 38 39 40 41 42
1 1 5 1 4 4 8 4 1 2
3 least connected regions:
66 71 79 with 1 link
2 most connected regions:
14 37 with 42 links
Weights style: B
Weights constants summary:
n nn S0 S1 S2
B 79 6241 1766 3532 226656Computing Gi statistics using Fixed Distance
fips <- order(boundary_with_accident_count$accident_count)
gi.fixed <- localG(boundary_with_accident_count$accident_count, wm62_lw)
gi.fixed [1] -3.35336397 -3.12912595 2.66840322 -3.40300611 -1.55553377 -1.60535680
[7] -3.49157140 -3.33512452 -2.30441253 1.39585137 2.02857406 -3.14674976
[13] -3.17369454 -3.11393349 -2.81776564 -2.72683627 -2.88374141 -3.09409303
[19] -3.12869158 -2.97426998 -1.24486361 -2.45871841 -0.73868252 -3.41408938
[25] -2.89311199 -3.10698762 -1.21501813 -3.40300611 -3.97875100 -3.49579305
[31] -3.68769474 0.40369815 -2.86023380 -2.45892382 -2.82195856 -0.92940375
[37] -3.00458192 -2.79839385 -3.02803442 -2.34863826 -2.70055541 -0.22060572
[43] -0.64115983 0.20224677 -3.19200023 1.40767563 -2.13429423 -2.10096499
[49] -2.78620527 -1.47043641 1.55439452 2.25397760 3.05249557 -3.47654632
[55] -0.01782564 4.36180044 -2.68397915 -2.21092597 0.65770388 -0.78671988
[61] -0.20919415 -1.51459191 -0.54373150 0.45951200 2.18029716 1.04197980
[67] -0.32317456 0.39447801 1.04576806 -0.64165535 -0.63894290 -0.13934645
[73] -0.31893930 -0.81293378 -1.00959192 0.67569728 -0.47712106 2.54803258
[79] 3.18220373
attr(,"internals")
Gi E(Gi) V(Gi) Z(Gi) Pr(z != E(Gi))
[1,] 0.272139227 0.51282051 0.0051513761 -3.35336397 7.983565e-04
[2,] 0.288233482 0.51282051 0.0051513761 -3.12912595 1.753271e-03
[3,] 0.136391769 0.05128205 0.0010173148 2.66840322 7.621273e-03
[4,] 0.255432270 0.50000000 0.0051650300 -3.40300611 6.664879e-04
[5,] 0.180171073 0.28205128 0.0042896392 -1.55553377 1.198190e-01
[6,] 0.333924168 0.44871795 0.0051132094 -1.60535680 1.084153e-01
[7,] 0.247971563 0.50000000 0.0052102341 -3.49157140 4.801881e-04
[8,] 0.273448329 0.51282051 0.0051513761 -3.33512452 8.526121e-04
[9,] 0.296456086 0.46153846 0.0051319328 -2.30441253 2.119950e-02
[10,] 0.241985442 0.16666667 0.0029115761 1.39585137 1.627593e-01
[11,] 0.224129353 0.12820513 0.0022360158 2.02857406 4.250169e-02
[12,] 0.258016812 0.48717949 0.0053035095 -3.14674976 1.650962e-03
[13,] 0.272139227 0.50000000 0.0051547652 -3.17369454 1.505120e-03
[14,] 0.313468692 0.53846154 0.0052205800 -3.11393349 1.846111e-03
[15,] 0.284889094 0.48717949 0.0051539569 -2.81776564 4.835909e-03
[16,] 0.278838749 0.47435897 0.0051412089 -2.72683627 6.394476e-03
[17,] 0.292251891 0.50000000 0.0051899304 -2.88374141 3.929813e-03
[18,] 0.290720062 0.51282051 0.0051526678 -3.09409303 1.974156e-03
[19,] 0.183188452 0.41025641 0.0052672775 -3.12869158 1.755865e-03
[20,] 0.261070466 0.47435897 0.0051424981 -2.97426998 2.936866e-03
[21,] 0.109258240 0.17948718 0.0031826485 -1.24486361 2.131819e-01
[22,] 0.260782502 0.43589744 0.0050725786 -2.45871841 1.394339e-02
[23,] 0.198059449 0.24358974 0.0037991399 -0.73868252 4.600998e-01
[24,] 0.203309875 0.44871795 0.0051668746 -3.41408938 6.399558e-04
[25,] 0.278947774 0.48717949 0.0051803995 -2.89311199 3.814453e-03
[26,] 0.302696456 0.52564103 0.0051489035 -3.10698762 1.890043e-03
[27,] 0.287300486 0.37179487 0.0048360434 -1.21501813 2.243592e-01
[28,] 0.255432270 0.50000000 0.0051650300 -3.40300611 6.664879e-04
[29,] 0.214054554 0.50000000 0.0051650300 -3.97875100 6.927825e-05
[30,] 0.219772515 0.47435897 0.0053037027 -3.49579305 4.726551e-04
[31,] 0.222076621 0.48717949 0.0051679573 -3.68769474 2.262949e-04
[32,] 0.348255907 0.32051282 0.0047227618 0.40369815 6.864347e-01
[33,] 0.268196569 0.47435897 0.0051953627 -2.86023380 4.233288e-03
[34,] 0.270124805 0.44871795 0.0052752059 -2.45892382 1.393542e-02
[35,] 0.244120853 0.44871795 0.0052565120 -2.82195856 4.773134e-03
[36,] 0.139284608 0.19230769 0.0032547762 -0.92940375 3.526799e-01
[37,] 0.322498456 0.53846154 0.0051664346 -3.00458192 2.659461e-03
[38,] 0.247349276 0.44871795 0.0051780488 -2.79839385 5.135745e-03
[39,] 0.282434515 0.50000000 0.0051624800 -3.02803442 2.461500e-03
[40,] 0.181588505 0.34615385 0.0049095816 -2.34863826 1.884220e-02
[41,] 0.138865303 0.32051282 0.0045243141 -2.70055541 6.922381e-03
[42,] 0.179613165 0.19230769 0.0033113077 -0.22060572 8.253994e-01
[43,] 0.251734311 0.29487179 0.0045266509 -0.64115983 5.214188e-01
[44,] 0.295476190 0.28205128 0.0044061520 0.20224677 8.397238e-01
[45,] 0.269990738 0.50000000 0.0051923603 -3.19200023 1.412912e-03
[46,] 0.228125000 0.15384615 0.0027843582 1.40767563 1.592271e-01
[47,] 0.246087091 0.39743590 0.0050286269 -2.13429423 3.281870e-02
[48,] 0.123152307 0.25641026 0.0040229869 -2.10096499 3.564404e-02
[49,] 0.144135802 0.33333333 0.0046111018 -2.78620527 5.332909e-03
[50,] 0.174530498 0.26923077 0.0041477249 -1.47043641 1.414436e-01
[51,] 0.137752297 0.07692308 0.0015314471 1.55439452 1.200903e-01
[52,] 0.078208290 0.02564103 0.0005439157 2.25397760 2.419758e-02
[53,] 0.259036145 0.11538462 0.0022146771 3.05249557 2.269471e-03
[54,] 0.169802258 0.42307692 0.0053074689 -3.47654632 5.079165e-04
[55,] 0.114558287 0.11538462 0.0021488926 -0.01782564 9.857780e-01
[56,] 0.193237100 0.05128205 0.0010591808 4.36180044 1.289965e-05
[57,] 0.205586331 0.39743590 0.0051093231 -2.68397915 7.275164e-03
[58,] 0.217455164 0.37179487 0.0048731182 -2.21092597 2.704096e-02
[59,] 0.174321909 0.14102564 0.0025628920 0.65770388 5.107284e-01
[60,] 0.067540006 0.10256410 0.0019819543 -0.78671988 4.314459e-01
[61,] 0.032644852 0.03846154 0.0007731295 -0.20919415 8.342967e-01
[62,] 0.104971080 0.19230769 0.0033250748 -1.51459191 1.298758e-01
[63,] 0.078550319 0.10256410 0.0019505280 -0.54373150 5.866262e-01
[64,] 0.036267748 0.02564103 0.0005348167 0.45951200 6.458665e-01
[65,] 0.100309499 0.03846154 0.0008046729 2.18029716 2.923544e-02
[66,] 0.029702206 0.01282051 0.0002624904 1.04197980 2.974210e-01
[67,] 0.052627500 0.06410256 0.0012607702 -0.32317456 7.465630e-01
[68,] 0.064076731 0.05128205 0.0010519939 0.39447801 6.932282e-01
[69,] 0.067548747 0.03846154 0.0007736300 1.04576806 2.956681e-01
[70,] 0.010752688 0.02564103 0.0005383798 -0.64165535 5.210970e-01
[71,] 0.002402356 0.01282051 0.0002658629 -0.63894290 5.228601e-01
[72,] 0.046816770 0.05128205 0.0010268471 -0.13934645 8.891764e-01
[73,] 0.041142416 0.05128205 0.0010107147 -0.31893930 7.497725e-01
[74,] 0.006984324 0.02564103 0.0005266952 -0.81293378 4.162560e-01
[75,] 0.018554918 0.05128205 0.0010508100 -1.00959192 3.126908e-01
[76,] 0.117669813 0.08974359 0.0017081266 0.67569728 4.992329e-01
[77,] 0.014813596 0.02564103 0.0005149838 -0.47712106 6.332759e-01
[78,] 0.175209043 0.07692308 0.0014878977 2.54803258 1.083324e-02
[79,] 0.064849332 0.01282051 0.0002673203 3.18220373 1.461590e-03
attr(,"cluster")
[1] Low Low Low Low Low Low Low Low Low Low High Low Low Low Low
[16] Low Low Low High Low High Low Low Low Low Low Low Low Low Low
[31] Low High Low High Low Low Low Low Low High Low Low High High Low
[46] High Low High Low Low High High High High Low High High Low Low High
[61] Low High Low High High Low Low High Low High Low Low Low Low High
[76] Low High Low Low
Levels: Low High
attr(,"gstari")
[1] FALSE
attr(,"call")
localG(x = boundary_with_accident_count$accident_count, listw = wm62_lw)
attr(,"class")
[1] "localG"The output of the localG() function is a vector containing G or G* values, with the following attributes: - "gstari": Indicates whether the G* version of the statistic was used (TRUE or FALSE). - "call": Stores the function call. - "class": Set to "localG", identifying the object type.
The Gi statistic is represented as a Z-score, where larger values signify stronger clustering. The sign of the value indicates the type of cluster: positive values point to high-value clusters (hotspots), while negative values indicate low-value clusters (cold spots).
To merge the Gi values with their corresponding geographic data in the Hunan spatial dataframe, use the following code to join the results to the boundary_with_accident_count sf object. This allows for the spatial visualization of clusters within the geographic data.
bmr.gi <- cbind(boundary_with_accident_count, as.matrix(gi.fixed)) %>%
rename(gstat_fixed = as.matrix.gi.fixed.)the code chunk above actually performs three tasks. First, it convert the output vector (i.e. gi.fixed) into r matrix object by using as.matrix(). Next, cbind() is used to join bmr@data and gi.fixed matrix to produce a new SpatialPolygonDataFrame called bmr.gi. Lastly, the field name of the gi values is renamed to gstat_fixed by using rename().
We can now map the Gi values derived using the fixed-distance weight matrix.
acc_count <- qtm(boundary_with_accident_count, "accident_count")
Gimap <-tm_shape(bmr.gi) +
tm_fill(col = "gstat_fixed",
style = "pretty",
palette="-RdBu",
title = "local Gi") +
tm_borders(alpha = 0.5)
tmap_arrange(acc_count, Gimap, asp=1, ncol=2)
From the above plot, we can infer that ‘hot spots’ tend to be neighboring regions and likewise for the cold spots too. We see more high value (hot) clusters in the South East region of the Bangkok Metropolitan Region- Near Samut Prakan. The Central Area, near Bangkok and Nonthaburi showcase more ‘cold spots’.
Conclusion
From all our analysis, we determine that the South East region of Bangkok Metropolitan Region seems to be a hotspot for accidents and needs to be further examined in order to reduce the rate of accidents in the region.